On the matching complexes of categorical product of path graphs

Raju Kumar Gupta
Department of Mathematics, Indian Institute of Technology (IIT) Delhi, India

Sourav Sarkar
Department of Mathematics, Indian Institute of Technology (IIT) Delhi, India

Sagar S. Sawant
Department of Mathematics, Indian Institute of Technology (IIT) Madras,  India

Samir Shukla
School of Mathematical and Statistical Sciences, Indian Institute of Technology (IIT) Mandi,  India

The matching complex $\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are  matchings in $G$. These complexes appears in various places and found applications in many areas of mathematics including; discrete geometry, representation theory, combinatorics, etc.
In this article, we consider the matching complexes of categorical product $P_n \times P_m$ of path graphs $P_n$ and $P_m$. For $m = 1$, $P_n \times P_m$ is a discrete graph and therefore its matching complex is  the void complex. For $m = 2$, $\mathsf{M}(P_n \times P_m)$ has been proved to be homotopy equivalent to a wedge of spheres by Kozlov. We show that for $n \geq 2$ and $3 \leq m \leq 5$, the matching complex of $P_n \times P_m$ is homotopy equivalent to a wedge of spheres. For $m =3$, we give a closed form formula for the number and dimension of spheres appearing in the wedge. Further, for $m \in \{4, 5\}$,  we give minimum and maximum dimension of spheres appearing in the wedge in the homotopy type of $\mathsf{M}(P_n \times P_m)$.

[2020]55P10, 05E45, 55U10

§ INTRODUCTION

A matching in a simple graph $G$ is a collection of pairwise disjoint edges. The matching complex $\M(G)$, of a graph $G$ is the simplicial complex whose vertices are the edges, and simplicies are the matchings in $G$. In topological combinatorics, it is an important problem to study the topology of matching complexes of graphs.  Matching complexes appeared first time in the thesis of Garst [7], where they obtained the matching complexes of complete bipartite graphs  $K_{m, n}$ as special coset complexes of symmetric groups $S_n$.  Thereafter, matching complexes appeared and found applications in various places.  The matching complexes of complete graphs appeared in the paper of Bouc  [5] in connection with Brown complexes and Quillen complexes.  In [4],  Björner et al., described the matching complexes of complete bipartite graphs as chessboard complexes to study the geometry of admissible rook configurations on a general $m\times n$ chessboard.  In [18],  $\M(K_{m, n})$ appeared as the complex of all partial, injective functions between $\{1, \ldots, m\}$ to $\{1, \ldots, n\}$ in connection to the problem of estimating the number of halving hyperplanes of a finite set of points in Euclidean space.
Other than the complete graphs and complete bipartite graphs, the classes of graphs for which matching complexes have been studied include; paths and cycles [12], trees [17, 16], polygonal tilings [15, 3], honeycomb graphs [17], etc.   For more details on these complexes, we refer to a survey article by Wachs [19] and Chapter 11 of the book [11] by Jonsson.

The cartesian product $G \Box H$ of graphs $G$ and $H$ is the graph with vertex set $V(G) \times  V(H)$ and any two vertices $(u, v)$  and $(u', v')$ form an edge if and only if either $u = u'$ and $vv' \in E(H)$, or $v = v'$ and $uu' \in E(G)$.

The categorical  product $G \times H$ of graphs $G$ and $H$ is the graph with vertex set $V(G) \times  V(H)$ and any two vertices $(u, v)$  and $(u', v')$ form an edge if and only if $uu' \in E(G)$ and $vv' \in E(H)$.

For $n \geq 1$, let  $P_n$ denote the path graph on $n$ vertices.  The matching complexes of the cartesian product of path graphs have been studied by many authors. In [12],  Kozlov studied  $\M(P_n \Box P_1)$.   In [10], Jonsson studied the homotopical depth and topological connectivity of matching complexes of the cartesian product of path graphs for certain cases. Further, the homology of $M(P_n \Box P_2)$  has been studied by Braun and Hough in [2], and their homotopy type by Matsushita in [14].  In [8], Goyal et al. showed that $\M(P_n \Box P_3)$ is homotopy equivalent to a wedge of spheres. For $m \geq 4$,  homotopy type of  $M(P_n \Box P_m)$ is not known.

Motivated from the cartesian product of path graphs, in this article, we consider the matching complexes of the categorical product  $P_n \times P_m$ of path graphs  (see <Ref>). Since  $P_n \times P_1$  does not contain any edge, $\M(P_n \times P_1)$ is a void complex. Further, $P_n \times P_2$ is a disjoint union of two path graphs, and therefore $\M(P_n \times P_2)$ is homotopy equivalent to a wedge of spheres from [12].
For $3 \leq m \leq 5$, we compute the homotopy type of the matching complexes of $P_n \times P_m$.

in 1,2,...,7

[vertex, label = -90:$1\x$] (1) at (,1) ;

in 1,2,...,7

[vertex, label = [label distance = -4pt]30:$2\x$] (2) at (,2) ;

in 1,2,...,7

[vertex, label = [label distance = -4pt]30:$3\x$] (3) at (,3) ;

in 1,2,...,7

[vertex, label = [label distance = -4pt]30:$4\x$] (4) at (,4) ;

in 1,2,...,7

[vertex, label = 90:$5\x$] (5) at (,5) ;

[evaluate = as using int(+1)] in 1,2,...,6

[edge] (1) – (1) ;
[edge] (2) – (2) ;
[edge] (3) – (3) ;
[edge] (4) – (4) ;
[edge] (5) – (5) ;

in 1,2,3,4,5,6,7

[evaluate = as using int(+1)] in 1,2,3,4

[edge] () – () ;

$P_7 \Box P_5$

in 1,3,5,7

[vertex, label = -90:$1\x$] (1) at (,1) ;

in 2,4,6

[vertex, label = 0:$2\x$] (2) at (,2) ;

in 1,3,5,7

[vertex, label = 0:$3\x$] (3) at (,3) ;

in 2,4,6

[vertex, label = 0:$4\x$] (4) at (,4) ;

in 1,3,5,7

[vertex, label = 90:$5\x$] (5) at (,5) ;

[evaluate = as using int(+1)] in 1,3,5

[edge] (5) – (4) ;
[edge] (3) – (2) ;

[edge] (1) – (2) ;
[edge] (3) – (4) ;

[evaluate = as using int(+1)] in 2,4,6

[edge] (4) – (5) ;
[edge] (2) – (3) ;

[edge] (2) – (1) ;
[edge] (4) – (3) ;

in 2,4,6

[vertex, label = -90:$1\x$] (1) at (,1) ;

in 1,3,5,7

[vertex, label = 0:$2\x$] (2) at (,2) ;

in 2,4,6

[vertex, label = 0:$3\x$] (3) at (,3) ;

in 1,3,5,7

[vertex, label = 0:$4\x$] (4) at (,4) ;

in 2,4,6

[vertex, label = 90:$5\x$] (5) at (,5) ;

[evaluate = as using int(+1)] in 2,4,6

[edge] (5) – (4) ;
[edge] (3) – (2) ;

[edge] (1) – (2) ;
[edge] (3) – (4) ;

[evaluate = as using int(+1)] in 1,3,5

[edge] (4) – (5) ;
[edge] (2) – (3) ;

[edge] (2) – (1) ;
[edge] (4) – (3) ;

$P_7 \times P_5$

Cartesian and categorical product of $P_7$ and $P_5$

For $n \geq 2$ and $3 \leq m \leq 5$, $\M(P_n \times P_m)$ is homotopy equivalent to a wedge of spheres.

We explicitly compute the dimension and number of spheres in the  homotopy type of $\M(P_n \times P_3)$ (see <Ref>). For $m \in \{4, 5\}$, we give the minimum and maximum dimension of spheres appearing in the wedge in homotopy type of $\M(P_n \times P_m)$.

For  $n \geq 3$, let $d_{max}M(P_n \times P_5)$ and $d_{min}M(P_n \times P_5)$, denote the maximum and minimum dimension of spheres appearing in the wedge in the homotopy type of $M(P_n \times P_5)$, respectively. Then $d_{max}M(P_n \times P_5)$ and $d_{min}M(P_n \times P_5)$ are given in <Ref>.

hline2-Z = 0.5pt, vline2-Z = 0.5pt,hline1 = 1-Z0.5pt,vline1 = 1-Z0.5pt,colspec=cccccc,r,cell11 = c=3c, cell14 = c=3c, cell71 = c=3c
$d_{max}M(P_n \times P_5)$                $d_{min}M(P_n \times P_5)$

$k \equiv$	    $n = 2k+2$       $n = 2k+3$     $k \equiv$      $n= 2k+2$     $n= 2k+3$

$0 \mod{4}$    $7(\frac{k}{2}) +1$      $7(\frac{k}{2}) +3$     $0 \mod{5}$      $16(\frac{k}{5})+1$        $16(\frac{k}{5})+4$

$1 \mod{4}$    $7(\frac{k-1}{2})+5$       $7(\frac{k-1}{2}) +7$      $1 \mod{5}$     $16(\frac{k-1}{5})+5$      $16(\frac{k-1}{5})+7$

$2 \mod{4}$    $7(\frac{k-2}{2})+9$       $7(\frac{k-2}{2}) +11$     $2\mod{5}$     $16(\frac{k-2}{5})+9$      $16(\frac{k-2}{5})+10$

$3 \mod{4}$    $7(\frac{k-3}{2})+13$     $7(\frac{k+1}{2})$      $3 \mod{5}$     $16(\frac{k-3}{5})+11$      $16(\frac{k-3}{5})+13$

$4 \mod{5}$     $16(\frac{k-4}{5})+15$      $16(\frac{k+1}{5})$

Maximum and minimum dimension of spheres in $M(P_n \times P_5)$

hline2-Z = 0.5pt, vline2-Z = 0.5pt,hline1 = 1-Z0.5pt,vline1 = 1-Z0.5pt,colspec=ccc,r
$k = 7 q+r$, $0 \leq r \leq 6$     $n = 2k+2$     $n = 2k+3$

$r = 0$     $2(k+3q)+1$     $2(k+3q)+1$

$r = 1$     $2(k+3q)+1$       $2(k+3q)+3$

$r = 2$     $2(k+3q)+3$       $2(k+3q)+3$

$r = 3, 4$     $2(k+3q)+3$      $2(k+3q)+5$

$r = 5, 6$     $2(k+3q)+5$      $2(k+3q)+7$

Maximum dimension of spheres in $M(P_n \times P_4)$

For $n \geq 3$, the minimum dimension of sphere in homotopy type of $M(P_n \times P_4)$ is $2k+1$ if $n = 2k+2$, and $2(k+1)+1$ if $n = 2k+3$. The maximum dimension of sphere in homotopy type of $M(P_n \times P_4)$  is given in <Ref>.

The line graph $L(G)$ of a graph $G$, is the graph whose vertices are the edges of $G$ and any two distinct vertices $uv$ and $u'v'$ joined by an edge if and only if they have a common endpoint, i.e., if $\{u, v\} \cap \{u', v'\}  \neq \emptyset$. For a graph $G$, a subset $A \subseteq V(G)$ is said to be independent  if the induced subgraph on $A$ does not contain any edge. The independence complex $\ind(G)$ of $G$ is a simplicial complex on $V(G)$ whose simplices are the independent sets of $G$.  Observe that $\M(G)$ is the same as the $\ind(L(G))$ for any graph $G$.

To prove <Ref>, we show that  $\ind(L(P_n \times P_m))$ is a wedge of spheres up to homotopy equivalence. The main idea used in this proof is to reduce the graph $L(P_n \times P_m)$ into smaller graphs by the operations; deletions of vertices, deletions of edges, and additions of edges.  During this process of reduction, we come across $13$ new classes of graphs in the case $m = 5$, and $4$ new classes of graphs in the case $m = 4$. We establish recursive relations among the independence complex of  $L(P_n \times P_m)$ and independence complexes of these new classes of graphs.

This paper is organized as follows: In <Ref>, we give necessary preliminaries related to graph theory and simplicial complexes, and results used in this article. <Ref> is majorly dedicated to the  matching complexes of  $P_n \times P_5$ and it is  divided into three subsections. The graph $L(P_n \times P_5)$ is disconnected and has exactly two components.  If $n = 2k+2$ is even, then  $L(P_n \times P_5)$ consists of two copies of the graph   $\Gamma_{k, 5}$ (see <Ref>) and if $n = 2k+3$ is odd,  then $L(P_n \times P_5) = \Lambda_{n, 5} \sqcup \tilde{\Gamma}_{n, 5}$ (see <Ref> and <Ref>).    We start   <Ref> with the definitions of the graphs  $\Gamma_{k, 5}, \Lambda_{k, 5}$ and $ \tilde{\Gamma}_{k, 5}$.  In <Ref>, we compute the independence complexes of $\Gamma_{k, 5}$ and $ \Lambda_{k, 5}$. We introduce  $6$ new classes of graphs (which we need in the computation of $\ind(\Gamma_{k, 5})$ and $\ind(\Lambda_{k, 5})$), and establish recursive relations among independence complexes of graphs $\Gamma_{k,5}, \Lambda_{k, 5}$ and these $6$ classes of graphs.
In <Ref>,  we compute independence complex of the graph $\tilde{\Gamma}_{k, 5}$. Here again,  we define another $7$ class of graphs (which we need in the computation of $\ind(\tilde{\Gamma}_{k, 5}))$, and establish recursive relations among independence complexes of these $7$ new classes of graphs and   $\ind(\tilde{\Gamma}_{k, 5})$.
In <Ref>, we conclude the homotopy type of $\M(P_n \times P_5)$ and determine the minimum and maximum dimensions of spheres appearing in the wedge in $\M(P_n \times P_5)$. In <Ref> and <Ref>, we compute the homotopy type of the matching complexes of $P_n \times P_4$ and $P_n \times P_3$, respectively. Finally, in  <Ref>, we pose a few questions and a conjecture, which naturally arise from the work done in this paper.

§ PRELIMINARIES

A graph $G$ is an ordered pair $(V(G),E(G))$, where $V(G)$ is a finite set, called the set of vertices and $E(G) \subseteq {V(G) \choose 2}$, the $2$-element subsets of $V(G)$,  called the set of edges of $G$. For the sake of brevity, we denote an edge $\{u,v\}$ by $uv$. Given a graph $G=(V(G),E(G))$ and a vertex $u$ in $V(G)$, the open neighborhood of $u$ is defined as the collection $\{v\in V(G) : uv\in E(G)\}$ and it is denoted  $N(u)$.  The closed neighborhood of a vertex $u$ in the graph $G$ is defined as $\{u\}\cup N(u)$ and it is denoted by $N[u]$. A map $f : V (G) \rightarrow V (H)$ is said to be a graph homomorphism if $f(v)f(w) \in E(H)$ for all $vw \in E(G)$. An isomorphism is a graph homomorphism that is bijective, with its inverse also being a graph homomorphism. If such an isomorphism exists between two graphs $G$ and $H$, they are called isomorphic, denoted as $G \cong H$. Let $S=\{v_1,\dots, v_r\}$ be a collection of vertices in $V(G)$. Then, by $G\setminus S$ we mean the induced subgraph of $G$ on the vertex set $V(G)\setminus S$. In particular, if $S$ is the singleton set $\{v_1\}$, then we denote the graph $G\setminus S$ by $G\setminus v_1$. Similarly, for a collection of edges $T=\{e_1,e_2,\dots,e_s\}$, by $G+T$ and $G-T$ we mean the graphs on the vertex set $V(G)$ with edge set $E(G)\cup T$ and $E(G)\setminus T$, respectively.  For more details about the graphs, one may refer to [20].

An (abstract) simplicial complex $\Delta$ on a vertex set $V$ is a collection of subsets of $V$ such that if $\sigma\in \Delta$ and $\tau$ is a subset of $\sigma$, then $\tau\in\Delta$. If $\sigma\in\Delta$ and $|\sigma|=k+1$, then we say $\sigma$ is a simplex of dimension $k$ or a $k$-simplex. We further assume that the empty set is present in every simplicial complex as the only simplex of dimension $-1$. The $0$-dimensional simplices are called the vertices of $\Delta$, and the set of all vertices of a simplicial complex $\Delta$ is denoted by $V(\Delta)$.

A subcomplex of a simplicial complex is a sub-collection of simplices that itself forms a simplicial complex. If $\Delta$ is a simplicial complex and $\sigma$ is a simplex in $\Delta$, then the link and the deletion of $\sigma$ is defined as the subcomplexes $\{\tau\in\Delta : \sigma\cup\tau\in\Delta \hspace{.2cm}\text{and}\hspace{.1cm} \sigma\cap\tau=\emptyset\}$ and $\{\tau\in\Delta : \tau\nsubseteq\sigma\}$, respectively.
The join of two simplicial complexes $\Delta_1$ and $\Delta_2$, where $V(\Delta_1)\cap V(\Delta_2)=\emptyset$, is the simplicial complex $\Delta_1\ast\Delta_2:=\{\sigma\sqcup\tau : \sigma\in\Delta_1\hspace{.2cm}\text{and}\hspace{.1cm}\tau\in\Delta_2\}$. In particular, if $a,b$ are vertices that are not part of a simplicial complex $\Delta$, then the simplicial complexes $\{a\}\ast\Delta$ and $\{a\}\ast\Delta\cup\{b\}\ast\Delta$ are called the cone and the suspension of $\Delta$ with the apexes $a$ and $\{a,b\}$, respectively. We denote the suspension of $\Delta$ with apex $\{a,b\}$ by $\Sigma_{a,b}\Delta$ (or $\Sigma\Delta$).

In this article,  we consider any simplicial
complex as a topological space, namely its geometric realization. For the definition of
geometric realization, we refer to [13]. For details about homotopy and topological background, one may refer to [9].

The following observation follows from the definition of the join of two simplicial complexes:

Let $G_1 \sqcup G_2$ denote the disjoint union of two graphs, $G_1$ and $G_2$. Then

$$\ind(G_1 \sqcup G_2) \simeq \ind(G_1) \ast \ind(G_2).$$

Clearly, for each vertex $v\in V(G)$, the link and the deletion of the simplex $\{v\}$ in $\ind(G)$ are the same as $\ind(G\setminus N[v])$ and $\ind(G\setminus v)$, respectively. Observe that $\ind(G)$ is the same as the simplicial complex $\{v\}\ast \ind(G\setminus N[v])\cup \ind(G\setminus v)$. Further, from [1], we have the following:

Let $v$ be a vertex of a graph $G$. If the inclusion $\ind(G\setminus \n[v]) \xhookrightarrow {} \ind(G \setminus v)$ is null-homotopic, then we have

$$\ind(G)\simeq \ind(G \setminus v) \vee \Sigma \ind(G\setminus N [v]).$$

The following propositions exhibit homotopy equivalence between the independence complex of a graph $G$ and the independence complexes of graphs derived from $G$ through certain graph operations involving the deletion of vertices, edges, or vertex-neighborhoods.

Let $v$ and $w$ be a pair of distinct vertices of $ G$ with $N(v) \subseteq N(w)$. Then
$$  \ind(G)\simeq {} \ind(G \setminus w) .$$

Let $G$ be a graph and $v$ be a simplicial vertex of $G$, that is, all neighbours of $v$ are pairwise adjacent. Let $N(v) = \{w_1,w_2,\dots,w_k\}$. Then

$$\ind(G) \simeq \bigvee_{i=1}^{k}\Sigma(\ind(G\setminus N [w_i])).$$

Let $G$ be a graph and let $uv$ be any edge in $G$. If $\ind(G\setminus \n[\{u, v\}])$ is contractible, then

$$\ind(G)\simeq \ind(G \setminus uv)$$

Based on the <Ref>, we have the following. Suppose $G$ is a graph that consists vertices $\{u,v\}$ and some vertex $x$ (depending on $u$ and $v$) such that $x \notin N(u) \cup N(v)$, and $N(x) \subseteq N(u) \cup N(v)$. Then $x$ is an isolated vertex in $G \setminus N[u,v]$, which implies $\ind(G\setminus N[\{u, v\}])$ is contractible. Thus, we have

\ind(G) \simeq \ind(G\triangle uv),

where $V(G \triangle uv) = V(G)$, and $E(G \triangle uv)$ is given by the symmetric difference $E(G) \triangle \{uv\}.$

A triplet of distinct vertices $u,v$ and $x$ in a graph $G$ satisfying $x \notin N(u) \cup N(v)$ and $N(x) \subseteq N(u) \cup N(v)$ is called an edge-invariant triplet in $G$, and denoted by $[u,v;x]$.

For $n \geq 1$, the cycle graph $\mathcal{C}_n$ is a graph with $V (\mathcal{C}_n) = \{v_i \mid 1\leq i\leq n\}$ and $E(\mathcal{C}_n) = \{v_iv_{i+1} \mid 1\leq i \leq n-1\}\cup \{v_1v_n\}$.

For $n \geq 3$,

$\ind(\mathcal{C}_n) \simeq \begin{cases}
\Sp^{k-1} \bigvee S^{k-1} & \text{if} \ n = 3k,\\
\Sp^{k-1} & \text{if} \ n = 3k \pm 1.\\
\end{cases}$

For $n \geq 1$, the path graph $P_n$ is a graph with $V (P_n) = \{v_i \mid 1\leq i\leq n\}$ and $E(P_n) = \{v_iv_{i+1} \mid  1\leq i \leq n-1\}$.

For $n \geq 1$,

$\ind(P_n) \simeq \begin{cases}
\Sp^{k-1}  & \text{if} \ n = 3k,\\
\text{a point} & \text{if} \ n = 3k+1,\\
\Sp^{k} & \text{if} \ n = 3k+2.\\
\end{cases}$

§ MATCHING COMPLEXES OF  MATCHING OF GRID

In this section, we compute the homotopy type of the matching complex of the categorical product $P_n \times P_5$. Since $P_n$ is bipartite, $P_n \times P_5$ (and its line graph) is disconnected and contains exactly two connected components. When $n$ is even, these components are isomorphic, whereas, for odd $n$, they are non-isomorphic.

For $n \geq 0$, we denote the connected component of the line graph of $P_{2n+2}\times P_5$ by $\Gamma_{n,5}$ (see <Ref>). Since the line graph  $L(P_{2n+2}\times P_5) = \Gamma_{n,5} \sqcup \Gamma_{n,5}$, the matching complex

\begin{equation*}
\M(P_{2n+2}\times P_5) \simeq \ind(L(P_{2n+2}\times P_5)) \simeq \ind(\Gamma_{n,5}) * \ind(\Gamma_{n,5}).
\end{equation*}

Similarly, for $n\geq 0$, let $\Lambda_{n,5}$ and $\Tilde{\Gamma}_{n,5}$ (see <Ref>) denote the two non-isomorphic connected components of the line graph of $P_{2n+3} \times P_5$. Therefore, we have

\begin{equation*}
\M(P_{2n+3}\times P_5) = \ind(L(P_{2n+3}\times P_5)) \simeq \ind(\Lambda_{n,5}) * \ind(\Tilde{\Gamma}_{n,5}).
\end{equation*}

We now define the graphs $\Gamma_{n, 5}, \Lambda_{n, 5}$ and $\Tilde{\Gamma}_{n, 5}$.

For $n\geq 0$, the graph $\Gamma_{n,5}$ comprises of the vertex set
$$V(\Gamma_{n,5}) = \set{g_{ij} \given i=1,2,3,4 \text{ and }j = {1,2,\dots,2n{+}1}},$$
where the vertices are arranged in $4$ rows and $2n+1$ columns. The edge set of $\Gamma_{n,5}$ is given by

\begin{align*}
E(\Gamma_{n,5}) = &\set{g_{ij}g_{i+1\,j} \given i=1,2,3, \text{ and } j=1,2,\dots,2n{+}1} \\
& \cup \set{g_{ij}g_{i\,j+1} \given i=1,2,3,4, \text{ and } j=1,2,\dots,2n} \\
& \cup \set{g_{ij}g_{i+1\,j+1}, g_{ij+1}g_{i+1\,j} \given i=1,2,3, \; j=1,2,\dots,2n, \text{ and } i+j\equiv 1 \mod{2}}.
\end{align*}

in 1,2,...,8

[vertex] (1) at (2*,6) ;

in 1,2,...,8

[vertex] (2) at (2*,2*2) ;

in 1,2,...,8

[vertex] (3) at (2*,1*2) ;

in 1,2,...,8

[vertex] (4) at (2*,0*2) ;

[evaluate=as using int(+1)] in 1,2,...,4,6,7

[edge] (1) to (1);
[edge] (2) – (2);
[edge] (3) – (3);
[edge] (4) – (4);

in 1,2,3,4,5,...,8
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{4\,{2n+1}}$] at (48) ;
[label=[label distance=-3pt]-90:$\cdots$] at (46) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

in 3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]180:$g_{21}$] at (21) ;
[label=[label distance=-3pt]180:$g_{31}$] at (31) ;
[label=[label distance=-3pt]180:$g_{41}$] at (41) ;;

[label=[label distance=-3pt]90:$g_{1\,{2n+1}}$] at (18) ;
[label=[label distance=-3pt]90:$\cdots$] at (16) ;
[label=[label distance=-3pt]0:$g_{2\,{2n+1}}$] at (28) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+1}}$] at (38) ;

[] at (9,-4) (b) $\Gamma_{n,5}$, for $n\geq 1$;

[shift = (-5,0)]
in 1,2,3,4
[vertex] (x1) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;

in 1,2,3,4

[label=[label distance=-3pt]0:$g_{\y 1}$] at (x1) ;

[] at (0,-4) (a) $\Gamma_{0,5}$;

For $n \geq 0$,  $\Lambda_{n,5}$ is a graph whose vertex set and edge set  is given by

\begin{align*}
V(\Lambda_{n,5}) = &\{l_1,l_2,l_3,l_4\}\cup V(\Gamma_{n,5}), \\
E(\Lambda_{n,5}) = & \{l_1l_2,l_2l_3,l_3l_4,g_{11}l_1,g_{11}l_2,g_{21}l_1,g_{21}l_2,g_{31}l_3,g_{31}l_4,g_{41}l_3,g_{41}l_4 \} \cup E(\Gamma_{n,5}).
\end{align*}

in 1,2,...,8

[vertex] (1) at (2*,3*2) ;

in 1,2,...,8

[vertex] (2) at (2*,2*2) ;

in 1,2,...,8

[vertex] (3) at (2*,1*2) ;

in 1,2,...,8

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2,...,4,6,7

[edge] () – ();

in 1,2,3,4,5,...,8
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 1,2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{4\,{2n+1}}$] at (48) ;
[label=[label distance=-3pt]-90:$\cdots$] at (46) ;
[label=[label distance=-3pt]0:$g_{2\,{2n+1}}$] at (28) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+1}}$] at (38) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

[label=[label distance = -8pt]190:$g_{21}$] at (21) ;
[label=[label distance = -8pt]-30:$g_{31}$] at (31) ;
in 3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]90:$g_{1\,{2n+1}}$] at (18) ;
[label=[label distance=-3pt]90:$\cdots$] at (16) ;

[vertex] (l1) at (-2,6)  ;
[vertex] (l2) at (-2,4)  ;
[vertex] (l3) at (-2,2)  ;
[vertex] (l4) at (-2,0)  ;

[edge] (l1) – (l2) ;
[edge] (l2) – (l3) ;
[edge] (l3) – (l4) ;
[edge] (l1) – (11) ;
[edge] (l2) – (21) ;
[edge] (l3) – (31) ;
[edge] (l4) – (41) ;
[edge] (l1) – (21) ;
[edge] (l2) – (11) ;
[edge] (l4) – (31) ;
[edge] (l3) – (41) ;

[label=[label distance=-3pt]180:$l_{1}$] at (l1) ;
[label=[label distance=-3pt]180:$l_{2}$] at (l2) ;
[label=[label distance=-3pt]180:$l_{3}$] at (l3) ;
[label=[label distance=-3pt]180:$l_{4}$] at (l4) ;

[] at (6,-4) (b) $\Lambda_{n,5}$, for $n\geq 1$;

[shift = (-6,0)]
in 1,2,3,4
[vertex] (x1) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;

[vertex] (l1) at (-2,6)  ;
[vertex] (l2) at (-2,4)  ;
[vertex] (l3) at (-2,2)  ;
[vertex] (l4) at (-2,0)  ;

[edge] (l1) – (l2) ;
[edge] (l2) – (l3) ;
[edge] (l3) – (l4) ;
[edge] (l1) – (x11) ;
[edge] (l2) – (x21) ;
[edge] (l3) – (x31) ;
[edge] (l4) – (x41) ;
[edge] (l1) – (x21) ;
[edge] (l2) – (x11) ;
[edge] (l4) – (x31) ;
[edge] (l3) – (x41) ;

in 1,2,3,4

[label=[label distance=-3pt]0:$g_{\y 1}$] at (x1) ;

[label=[label distance=-3pt]180:$l_{1}$] at (l1) ;
[label=[label distance=-3pt]180:$l_{2}$] at (l2) ;
[label=[label distance=-3pt]180:$l_{3}$] at (l3) ;
[label=[label distance=-3pt]180:$l_{4}$] at (l4) ;

[] at (-1,-4) (a) $\Lambda_{0,5}$;

$\Lambda_{n, 5}$

For $n \geq 0$, $\tilde{\Gamma}_{n, 5}$ is a graph whose  vertex set and edge set is given by

\begin{equation*}
\begin{split}
V(\Tilde{\Gamma}_{n,5}) =  & \ V(\Gamma_{n,5})\cup \{g_{1\,2n+2},g_{2\,2n+2},g_{3\,2n+2},g_{4\,2n+2}\} \\
E(\Tilde{\Gamma}_{n,5}) = & \ E(\Gamma_{n,5}) \cup \set{g_{i\,2n+1}g_{i\,2n+2} \given i=1,2,3,4} \\
& \cup \set{g_{i\,2n+2}g_{i+1\,2n+2} \given i = 1,2,3} \\
& \cup \set{g_{2\,2n+1}g_{3\,2n+2}, g_{3\,2n+1}g_{2\,2n+2}}.
\end{split}
\end{equation*}

in 1,2,...,9

[vertex] (1) at (2*,3*2) ;

in 1,2,...,9

[vertex] (2) at (2*,2*2) ;

in 1,2,...,9

[vertex] (3) at (2*,1*2) ;

in 1,2,...,9

[vertex] (4) at (2*,0*2) ;

[evaluate=as using int(+1)] in 1,2,...,4,6,7,8

[edge] (1) – (1);
[edge] (2) – (2);
[edge] (3) – (3);
[edge] (4) – (4);

in 1,2,3,4,5,...,8,9
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6,8

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7,9

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

in 3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]180:$g_{21}$] at (21) ;
[label=[label distance=-3pt]180:$g_{31}$] at (31) ;
[label=[label distance=-3pt]180:$g_{41}$] at (41) ;;
[label=[label distance=-3pt]90:$g_{1\,{2n+2}}$] at (19) ;
[label=[label distance=-3pt]90:$\cdots\cdots$] at (16) ;

[label=[label distance=-3pt]0:$g_{2\,{2n+2}}$] at (29) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+2}}$] at (39) ;
[label=[label distance=-3pt]0:$g_{4\,{2n+2}}$] at (49) ;

[] at (9,-4) (b) $\Tilde{\Gamma}_{n,5}$, for $n\geq 1$;

in 1,2,3,4
[vertex] (x1) at (-2,2*(4-) ;
[vertex] (x2) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;
[edge] (x2) – (x2) ;

[edge] (x11) – (x12) ;
[edge] (x21) – (x22) ;
[edge] (x21) – (x32) ;
[edge] (x31) – (x22) ;
[edge] (x31) – (x32) ;
[edge] (x41) – (x42) ;

in 1,2,3,4

[label=[label distance=-3pt]180:$g_{\y 1}$] at (x1) ;
[label=[label distance=-3pt]0:$g_{\y 2}$] at (x2) ;

[] at (-1,-4) (a) $\Tilde{\Gamma}_{0,5}$;

In the following section,  we compute the homotopy type of $\ind(\Gamma_{n, 5})$ and $\ind(\Lambda_{n, 5})$, and the homotopy type of $\ind(\tilde{\Gamma}_{n, 5})$ is computed in <Ref>. Finally, we give the minimum and maximum dimension of spheres in the homotopy type of $\M(P_n \times P_5)$ in <Ref>.

§.§ Independence complexes of  $\Gamma_{n,5}$ and $\Lambda_{n,5}$

In this section, we introduce $6$ new classes of graphs, namely; $A_{n,5}, B_{n,5}, C_{n,5}, D_{n,5},$ $E_{n,5}$, and $F_{n,5}$ for $n \geq 0$. We show that the independence complexes of $\Gamma_{n,5}$ and $\Lambda_{n,5}$ can be expressed in terms of the wedge of suspensions of independence complexes of these intermediary graphs. Through induction on $n$, we establish that the independence complexes of $\Gamma_{n,5}$, $\Lambda_{n,5}$, and these intermediary graphs are homotopy equivalent to a wedge of spheres.

We begin with the description of the intermediary graphs and compute the homotopy type of their independence complexes for $n=0$.

§.§.§ Definitions of intermediary graphs and their independence complexes for $n = 0$

(i) Graph $A_{n,5}$.
For $n\geq 0$, we define the graph $A_{n,5}$ as follows:

\begin{equation*}
\begin{split}
V(A_{n,5})=& V(\Gamma_{n,5})\cup \{a_1,a_2\},\\
E(A_{n,5})=& E(\Gamma_{n,5})\cup \{a_{1}g_{11},a_{1}g_{21},a_{2}g_{31},a_{2}g_{41}\}.
\end{split}
\end{equation*}

in 1,2,...,8

[vertex] (1) at (2*,3*2) ;

in 1,2,...,8

[vertex] (2) at (2*,2*2) ;

in 1,2,...,8

[vertex] (3) at (2*,1*2) ;

in 1,2,...,8

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2,...,4,6,7

[edge] () – ();

in 1,2,3,4,5,...,8
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 1,2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{4\,{2n+1}}$] at (48) ;
[label=[label distance=-3pt]-90:$\cdots$] at (46) ;
[label=[label distance=-3pt]0:$g_{2\,{2n+1}}$] at (28) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+1}}$] at (38) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

in 3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[vertex] (a1) at (-2,6)  ;
[vertex] (a2) at (-2,0)  ;

[edge] (a1) – (11) ;
[edge] (a1) – (21) ;
[edge] (a2) – (31) ;
[edge] (a2) – (41) ;

[label=[label distance=-3pt]180:$a_{1}$] at (a1) ;
[label=[label distance=-3pt]180:$a_{2}$] at (a2) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]90:$g_{1\,{2n+1}}$] at (18) ;
[label=[label distance=-3pt]90:$\cdots$] at (16) ;
in 2,3

[label=[label distance=-3pt]180:$g_{\x1}$] at (1) ;

[] at (6,-4) (b) $A_{n,5}$, for $n\geq 1$;

[shift = (-6,0)]
in 1,2,3,4
[vertex] (x1) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;

[vertex] (a1) at (-2,6)  ;
[vertex] (a2) at (-2,0)  ;

[edge] (a1) – (x11) ;
[edge] (a1) – (x21) ;
[edge] (a2) – (x31) ;
[edge] (a2) – (x41) ;

in 1,2,3,4

[label=[label distance=-3pt]0:$g_{\y 1}$] at (x1) ;

[label=[label distance=-3pt]180:$a_{1}$] at (a1) ;
[label=[label distance=-3pt]180:$a_{2}$] at (a2) ;

[] at (-1,-4) (a) $A_{0,5}$;

$\ind(A_{0,5}) \simeq \Sp^1 \vee \Sp^1 \vee \Sp^1$.

Since $a_1$ is a simplicial vertex in the graph $A_{0,5}$ with $N(a_1)=\{g_{11},g_{21}\}$, using <Ref> we get

\begin{equation*}
\ind(A_{0,5}) \simeq \Sigma\ind(A_{0,5} \setminus \n[g_{11}]) \vee \Sigma\ind(A_{0,5} \setminus \n[g_{21}]).
\end{equation*}

However, $A_{0,5} \setminus \n[g_{11}]$ is isomorphic to $K_3$, and $A_{0,5} \setminus \n[g_{21}]$ is isomorphic to $K_2$. Therefore, $\ind(A_{0,5} \setminus \n[g_{11}])\simeq \mathbb{S}^0\vee \mathbb{S}^0$ and $\ind(A_{0,5} \setminus \n[g_{21}])\simeq \mathbb{S}^0$. Hence, we have the stated conclusion.

(ii) Graph $B_{n,5}$.
For $n\geq 0$, we define the graph $B_{n,5}$ as follows:

\begin{equation*}
\begin{split}
V(B_{n,5})=& V(\Gamma_{n,5})\cup \{b_1,b_2, b_3,b_4,b_5,b_6\},\\
E(B_{n,5})=& E(\Gamma_{n,5})\cup \{b_{1}b_{2}, b_{1}b_{4},b_{1}b_{5},b_{2}b_{6},b_{3}b_{4},b_{4}b_{5},b_{5}b_{6}\}\\
& \cup \{b_{3}g_{11},b_{3}g_{21},b_{4}g_{11},b_{4}g_{21},b_{5}g_{31},b_{5}g_{41},b_{6}g_{31},b_{6}g_{41}\}.
\end{split}
\end{equation*}

Furthermore, we define $B_{-1}$ as a graph on vertex set $\{b_1,b_2\}$, and edge set $\{b_1b_2\}$. Thus, $B_{-1} \cong K_2$.

in 1,2,...,8

[vertex] (1) at (2*,3*2) ;

in 1,2,...,8

[vertex] (2) at (2*,2*2) ;

in 1,2,...,8

[vertex] (3) at (2*,1*2) ;

in 1,2,...,8

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2,...,4,6,7

[edge] () – ();

in 1,2,3,4,5,...,8
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 1,2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{4\,{2n+1}}$] at (48) ;
[label=[label distance=-3pt]-90:$\cdots$] at (46) ;
[label=[label distance=-3pt]0:$g_{2\,{2n+1}}$] at (28) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+1}}$] at (38) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

in 1,3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[vertex] (b1) at (-2,6)  ;
[vertex] (b2) at (-2,4)  ;
[vertex] (b3) at (-2,2)  ;
[vertex] (b4) at (-2,0)  ;
[vertex] (b31) at (-4,2)  ;
[vertex] (b41) at (-4,0)  ;

[edge] (b1) – (11) ;
[edge] (b1) – (21) ;
[edge] (b1) – (b2) ;
[edge] (b2) – (b3) ;
[edge] (b2) – (21) ;
[edge] (b2) – (11) ;
[edge] (b3) – (31) ;
[edge] (b3) – (41) ;
[edge] (b3) – (b4) ;
[edge] (b4) – (41) ;
[edge] (b4) – (31) ;

[edge] (b31) – (b3) ;
[edge] (b31) – (b2) ;
[edge] (b31) – (b41) ;
[edge] (b41) – (b4) ;

[label=[label distance=-3pt]180:$b_{3}$] at (b1) ;
[label=[label distance=-3pt]180:$b_{4}$] at (b2) ;
[label=[label distance=-7pt]220:$b_{5}$] at (b3) ;
[label=[label distance=-3pt]270:$b_{6}$] at (b4) ;
[label=[label distance=-3pt]180:$b_{1}$] at (b31) ;
[label=[label distance=-3pt]180:$b_{2}$] at (b41) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]90:$g_{1\,{2n+1}}$] at (18) ;
[label=[label distance=-3pt]90:$\cdots$] at (16) ;

[] at (5,-4) (b) $B_{n,5}$, for $n\geq 1$;

[shift = (-8,0)]
in 1,2,3,4
[vertex] (x1) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;

[vertex] (b1) at (-2,6)  ;
[vertex] (b2) at (-2,4)  ;
[vertex] (b3) at (-2,2)  ;
[vertex] (b4) at (-2,0)  ;
[vertex] (b31) at (-4,2)  ;
[vertex] (b41) at (-4,0)  ;

[edge] (b1) – (x11) ;
[edge] (b1) – (x21) ;
[edge] (b1) – (b2) ;
[edge] (b2) – (b3) ;
[edge] (b2) – (x21) ;
[edge] (b2) – (x11) ;
[edge] (b3) – (x31) ;
[edge] (b3) – (x41) ;
[edge] (b3) – (b4) ;
[edge] (b4) – (x41) ;
[edge] (b4) – (x31) ;

[edge] (b31) – (b3) ;
[edge] (b31) – (b2) ;
[edge] (b31) – (b41) ;
[edge] (b41) – (b4) ;

[label=[label distance=-3pt]180:$b_{3}$] at (b1) ;
[label=[label distance=-3pt]180:$b_{4}$] at (b2) ;
[label=[label distance=-7pt]220:$b_{5}$] at (b3) ;
[label=[label distance=-3pt]270:$b_{6}$] at (b4) ;
[label=[label distance=-3pt]180:$b_{1}$] at (b31) ;
[label=[label distance=-3pt]180:$b_{2}$] at (b41) ;

in 1,2,3,4

[label=[label distance=-3pt]0:$g_{\y 1}$] at (x1) ;

[] at (-2,-4) (a) $B_{0,5}$;

$\ind(B_{0,5}) \simeq \Sp^1 \vee \Sp^2$.

Since $g_{41}$ is a simplicial vertex in graph $B_{0,5}$ with $N(g_{41})=\{g_{31},b_5,b_{6}\}$, using <Ref> we get
\begin{equation*}
\ind(B_{0,5}) \simeq \Sigma(\ind(B_{0,5}\setminus \n[g_{31}]))\vee \Sigma(\ind(B_{0,5} \setminus \n[b_{6}]))\vee \Sigma(\ind(B_{0,5}\setminus \n[b_5])) .
\end{equation*}

Now, $b_2$ is an isolated vertex in the graph $B_{0,5}\setminus \n[b_5]$. Therefore,  $\ind(B_{0,5}\setminus \n[b_5])$ is contractible. Since $\n(b_2)\subseteq \n(b_4)$ in the graph $B_{0,5}\setminus \n[g_{31}]$, using <Ref> we get $\ind(B_{0,5}\setminus \n[g_{31}])\simeq \ind(B_{0,5}\setminus \{\n[g_{31}]\cup \{b_4\}\})$. Since the graph $B_{0,5}\setminus \{\n[g_{31}]\cup \{b_4\}\}$ consists of two disjoint edges, we have $\ind(B_{0,5}\setminus \n[g_{31}])\simeq\ind(K_2)\ast\ind(K_2)$, i.e., $\ind(B_{0,5}\setminus \n[g_{31}])\simeq \mathbb{S}^1$. Similarly, $\n(b_1)\subseteq\n(b_3)\cap\n(g_{11})\cap \n(g_{21})$ in the graph  $B_{0,5} \setminus \n[b_{6}]$ implies that $\ind(B_{0,5} \setminus \n[b_{6}])\simeq \ind(B_{0,5} \setminus \{\n[b_{6}]\cup\{b_3,g_{11},g_{21}\}\})$. Since $B_{0,5} \setminus \{\n[b_{6}]\cup\{b_3,g_{11},g_{21}\}\}$ is isomorphic to the graph $K_2$, we have $\ind(B_{0,5} \setminus \n[b_{6}])\simeq \mathbb{S}^0$. Therefore, using the fact that $\ind(B_{0,5}\setminus \n[g_{31}])\simeq \mathbb{S}^1$ and $\ind(B_{0,5} \setminus \n[b_{6}])\simeq\mathbb{S}^0$, we get $\ind(B_{0,5}) \simeq \Sp^1 \vee \Sp^2$.

(iii) Graph $C_{n,5}$.
For $n\geq 0$, we define the graph $C_{n,5}$ as follows:

\begin{equation*}
\begin{split}
V(C_{n,5})=&V(\Gamma_{n,5})\cup \{c_1\},\\
E(C_{n,5})=&E(\Gamma_{n,5})\cup \{c_{1}g_{31},c_{1}g_{41}\}.
\end{split}
\end{equation*}

in 1,2,...,8

[vertex] (1) at (2*,3*2) ;

in 1,2,...,8

[vertex] (2) at (2*,2*2) ;

in 1,2,...,8

[vertex] (3) at (2*,1*2) ;

in 1,2,...,8

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2,...,4,6,7

[edge] () – ();

in 1,2,3,4,5,...,8
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 1,2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{4\,{2n+1}}$] at (48) ;
[label=[label distance=-3pt]-90:$\cdots$] at (46) ;
[label=[label distance=-3pt]0:$g_{2\,{2n+1}}$] at (28) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+1}}$] at (38) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

in 3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]90:$g_{1\,{2n+1}}$] at (18) ;
[label=[label distance=-3pt]90:$\cdots$] at (16) ;
in 2,3

[label=[label distance=-3pt]180:$g_{\x1}$] at (1) ;

[] at (6,-4) (b) $C_{n,5}$, for $n\geq 1$;

[vertex] (c2) at (-2,0)  ;

[edge] (c2) – (31) ;
[edge] (c2) – (41) ;

[label=[label distance=-3pt]180:$c_{1}$] at (c2) ;

[shift = (-5,0)]
in 1,2,3,4
[vertex] (x1) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;

[vertex] (c2) at (-2,0)  ;

[edge] (c2) – (x31) ;
[edge] (c2) – (x41) ;

in 1,2,3,4

[label=[label distance=-3pt]0:$g_{\y 1}$] at (x1) ;

[label=[label distance=-3pt]180:$c_{1}$] at (c2) ;

[] at (-1,-4) (a) $C_{0,5}$;

$\ind(C_{0,5}) \simeq \Sp^1$.

Since $N(g_{11})\subseteq N(g_{31})$ in the graph $C_{0,5}$, using <Ref>, we conclude that $\ind(C_{0,5})\simeq\ind(C_{0,5}\setminus\{g_{31}\})$. However, the graph $C_{0,5}\setminus\{g_{31}\}$ consists of two disjoint edges. Therefore, $\ind(C_{0,5})\simeq \mathbb{S}^1.$

(iv) Graph $D_{n,5}$.
For $n\geq 0$, we define the graph $D_{n,5}$ as follows:

\begin{equation*}
\begin{split}
V(D_{n,5})= & V(\Gamma_{n,5})\cup \{d_{1},d_{2}\},\\
E(D_{n,5})= & E(\Gamma_{n,5})\cup \{d_{1}d_{2},d_{1}g_{11},d_{1}g_{21},d_{2}g_{31},d_{2}g_{41}\}.
\end{split}
\end{equation*}

in 1,2,...,8

[vertex] (1) at (2*,3*2) ;

in 1,2,...,8

[vertex] (2) at (2*,2*2) ;

in 1,2,...,8

[vertex] (3) at (2*,1*2) ;

in 1,2,...,8

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2,...,4,6,7

[edge] () – ();

in 1,2,3,4,5,...,8
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 1,2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{4\,{2n+1}}$] at (48) ;
[label=[label distance=-3pt]-90:$\cdots$] at (46) ;
[label=[label distance=-3pt]0:$g_{2\,{2n+1}}$] at (28) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+1}}$] at (38) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

in 1,3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]90:$g_{1\,{2n+1}}$] at (18) ;
[label=[label distance=-3pt]90:$\cdots$] at (16) ;

[vertex] (d1) at (-2,4)  ;
[vertex] (d2) at (-2,2)  ;

[edge] (d1) – (11) ;
[edge] (d1) – (21) ;
[edge] (d1) – (d2) ;
[edge] (d2) – (31) ;
[edge] (d2) – (41) ;

[label=[label distance=-3pt]180:$d_{1}$] at (d1) ;
[label=[label distance=-3pt]180:$d_{2}$] at (d2) ;

[] at (6,-4) (b) $D_{n,5}$, for $n\geq 1$;

[shift = (-5,0)]
in 1,2,3,4
[vertex] (x1) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;

[vertex] (d1) at (-2,4)  ;
[vertex] (d2) at (-2,2)  ;

[edge] (d1) – (x11) ;
[edge] (d1) – (x21) ;
[edge] (d1) – (d2) ;
[edge] (d2) – (x31) ;
[edge] (d2) – (x41) ;

in 1,2,3,4

[label=[label distance=-3pt]0:$g_{\y 1}$] at (x1) ;

[label=[label distance=-3pt]180:$d_{1}$] at (d1) ;
[label=[label distance=-3pt]180:$d_{2}$] at (d2) ;

[] at (-1,-4) (a) $D_{0,5}$;

$\ind(D_{0,5}) \simeq \Sp^1\vee \Sp^1$.

Since  $g_{11}$ is a simplicial vertex in the graph $D_{0,5}$ with $N(g_{11})=\{g_{21},d_1\}$, using  <Ref>, we get
$\ind (D_{0,5})\simeq\Sigma(\ind(D_{0,5} \setminus \n[g_{21}])) \vee \Sigma(\ind(D_{0,5} \setminus \n[d_1])).$
However, both the graphs $D_{0,5} \setminus \n[g_{21}]$, and $D_{0,5} \setminus \n[d_1]$ are isomorphic to the complete graph $K_2$. Therefore, both  $\ind(D_{0,5} \setminus \n[g_{21}])$ and  $\ind(D_{0,5} \setminus \n[d_1])$ are homotopy equivalent to $\mathbb{S}^0$. Hence the result.

(v) Graph $E_{n,5}$.
For $n\geq 0$, we define the graph $E_{n,5}$ as follows:

\begin{equation*}
\begin{split}
V(E_{n,5})=&V(\Gamma_{n,5})\cup \{e_{1}\},\\
E(E_{n,5})=&E(\Gamma_{n,5})\cup \{e_{1}g_{11},e_{1}g_{21},e_{1}g_{31},e_{1}g_{41}\}.
\end{split}
\end{equation*}

in 1,2,...,8

[vertex] (1) at (2*,3*2) ;

in 1,2,...,8

[vertex] (2) at (2*,2*2) ;

in 1,2,...,8

[vertex] (3) at (2*,1*2) ;

in 1,2,...,8

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2,...,4,6,7

[edge] () – ();

in 1,2,3,4,5,...,8
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 1,2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{4\,{2n+1}}$] at (48) ;
[label=[label distance=-3pt]-90:$\cdots$] at (46) ;
[label=[label distance=-3pt]0:$g_{2\,{2n+1}}$] at (28) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+1}}$] at (38) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

in 1,3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]90:$g_{1\,{2n+1}}$] at (18) ;
[label=[label distance=-3pt]90:$\cdots$] at (16) ;

[] at (6,-4) (b) $E_{n,5}$, for $n\geq 1$;

[vertex] (e1) at (-2,3)  ;

[edge] (e1) – (11) ;
[edge] (e1) – (21) ;
[edge] (e1) – (31) ;
[edge] (e1) – (41) ;

[label=[label distance=-3pt]180:$e_{1}$] at (e1) ;

[shift = (-5,0)]
in 1,2,3,4
[vertex] (x1) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;

[vertex] (e1) at (-2,3)  ;

[edge] (e1) – (x11) ;
[edge] (e1) – (x21) ;
[edge] (e1) – (x31) ;
[edge] (e1) – (x41) ;

[label=[label distance=-3pt]180:$e_{1}$] at (e1) ;
in 1,2,3,4

[label=[label distance=-3pt]0:$g_{\y 1}$] at (x1) ;

[] at (-1,-4) (a) $E_{0,5}$;

$\ind(E_{0,5}) \simeq \mathbb{S}^0$.

Since $N(g_{11})\subseteq N(g_{31})$ in the graph $E_{0,5}$, by <Ref>, we get $\ind (E_{0,5})\simeq \ind (E_{0,5}\setminus g_{31}) $. Furthermore, $N(g_{41})\subseteq N(g_{11})$ in $E_{n,5} \setminus g_{31}$, and an application of <Ref> provides us with $\ind (E_{0,5})\simeq \ind (E_{0,5}\setminus \{g_{11},g_{31}\})$. Note that the graph $E_{0,5}\setminus \{g_{11},g_{31}\}$ is a path on three vertices. Therefore, $ \ind (E_{0,5})\simeq \mathbb{S}^0 $.

(vi) Graph $F_{n,5}$.
For $n\geq 0$, we define the graph $F_{n,5}$ as follows:

\begin{equation*}
\begin{split}
V(F_{n,5})=& V(\Gamma_{n,5})\cup \{f_1,f_2, f_3\},\\
E(F_{n,5})=& E(\Gamma_{n,5})\cup \{f_{1}f_{2}, f_{2}f_{3},f_{1}g_{11},f_{1}g_{21},f_{2}g_{31},f_{2}g_{41},f_{3}g_{31},f_{3}g_{41}\}.
\end{split}
\end{equation*}

in 1,2,...,8

[vertex] (1) at (2*,3*2) ;

in 1,2,...,8

[vertex] (2) at (2*,2*2) ;

in 1,2,...,8

[vertex] (3) at (2*,1*2) ;

in 1,2,...,8

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2,...,4,6,7

[edge] () – ();

in 1,2,3,4,5,...,8
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2,4,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3,5,8

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1,3,6

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4,7

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[decoration=brace,raise=5pt,mirror,decorate, below]
(41 |- 52,-0.8) – node[below=6pt] $1$(43|- 0,-0.8);
(43|- 0,-0.8) – node[below=6pt] $2$(45|- 0,-0.8);
(46|- 0,-0.8) – node[below=6pt] $n$(48|- 0,-0.8);

in 1,2,3,4,5

[label=[label distance = -3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{4\,{2n+1}}$] at (48) ;
[label=[label distance=-3pt]-90:$\cdots$] at (46) ;
[label=[label distance=-3pt]0:$g_{2\,{2n+1}}$] at (28) ;
[label=[label distance=-3pt]0:$g_{3\,{2n+1}}$] at (38) ;

in 2,4

[label=[label distance = -8pt]-30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]30:$g_{3\x}$] at (3) ;

in 1,3,5

[label=[label distance = -8pt]30:$g_{2\x}$] at (2) ;
[label=[label distance = -8pt]-30:$g_{3\x}$] at (3) ;

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-3pt]90:$g_{1\,{2n+1}}$] at (18) ;
[label=[label distance=-3pt]90:$\cdots$] at (16) ;

[] at (6,-4) (b) $F_{n,5}$, for $n\geq 1$;

[vertex] (f2) at (-2,4)  ;
[vertex] (f3) at (-2,2)  ;
[vertex] (f4) at (-2,0)  ;

[edge] (f2) – (f3) ;
[edge] (f3) – (f4) ;
[edge] (f2) – (21) ;
[edge] (f3) – (31) ;
[edge] (f4) – (41) ;
[edge] (f2) – (11) ;
[edge] (f4) – (31) ;
[edge] (f3) – (41) ;

[label=[label distance=-3pt]180:$f_{1}$] at (f2) ;
[label=[label distance=-3pt]180:$f_{2}$] at (f3) ;
[label=[label distance=-3pt]180:$f_{3}$] at (f4) ;

[shift = (-5,0)]
in 1,2,3,4
[vertex] (x1) at (0,2*(4-) ;

[evaluate = as using int(+1)] in 1,2,3
[edge] (x1) – (x1) ;

[vertex] (f2) at (-2,4)  ;
[vertex] (f3) at (-2,2)  ;
[vertex] (f4) at (-2,0)  ;

[edge] (f2) – (f3) ;
[edge] (f3) – (f4) ;
[edge] (f2) – (x21) ;
[edge] (f3) – (x31) ;
[edge] (f4) – (x41) ;
[edge] (f2) – (x11) ;
[edge] (f4) – (x31) ;
[edge] (f3) – (x41) ;

[label=[label distance=-3pt]180:$f_{1}$] at (f2) ;
[label=[label distance=-3pt]180:$f_{2}$] at (f3) ;
[label=[label distance=-3pt]180:$f_{3}$] at (f4) ;

in 1,2,3,4

[label=[label distance=-3pt]0:$g_{\y 1}$] at (x1) ;

[] at (-1,-4) (a) $F_{0,5}$;

$\ind(F_{0,5}) \simeq \vee_{4}\Sp^1$.

Since   $g_{11}$ is a simplicial vertex in $E_{0,5}$ with $N(g_{11})=\{g_{21},f_{1}\}$, using <Ref> we get
$\ind (F_{0,5})\simeq\Sigma(\ind(F_{0,5} \setminus \n[g_{21}])) \vee \Sigma(\ind(F_{0,5} \setminus \n[f_1])).$
Since both the graphs $F_{0,5} \setminus \n[g_{21}]$ and $F_{0,5} \setminus \n[f_1]$ are isomorphic to the complete graph $K_3$, the result follows.

§.§.§ General case computation

In this section,  we establish recursive relations among the independence complexes of the graphs in
$\{\Gamma_{n,5},\Lambda_{n,5},A_{n,5},B_{n,5},C_{n,5},D_{n,5},E_{n,5}, $ $F_{n,5}\}$.
We proceed by induction on $n$ to show that the homotopy type of independence complexes of $\Gamma_{n,5}, \Lambda_{n,5}$ and other intermediary graphs is a wedge of spheres. We have proved in <Ref> that independence complexes of graphs in $\{\Gamma_{0,5},\Lambda_{0,5},A_{0,5},B_{0,5}, $ $C_{0,5},D_{0,5},E_{0,5},F_{0,5}\}$ are homotopy equivalent to wedge of spheres. By induction hypothesis, assume that for all $k < n$, the independence complexes of graphs in $\{\Gamma_{k,5},\Lambda_{k,5},A_{k,5}, $ $B_{k,5},C_{k,5},D_{k,5},E_{k,5},F_{k,5}\}$ are wedge of spheres.
We conclude the proof for $n$ by induction and the recurrence relations among independence complexes of these graphs.
For $n\geq 1$,

\begin{equation*}
\small
\ind(\Gamma_{n,5}) \simeq \begin{cases}
\bigvee_3\mathbb{S}^2 & \text{if $n = 1$,} \\
\bigvee_7\mathbb{S}^4 & \text{if $n = 2$,} \\
\Sigma^{3}\ind(A_{n-2,5}) \bigvee \Sigma^4 \ind(B_{n-3,5}) \bigvee \Sigma^{3} \ind(C_{n-2,5}) \bigvee \Sigma^{3} \ind(D_{n-2,5}) & \text{if $n \geq 3$.}
\end{cases}
\end{equation*}

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 3

[vertex] (2) at (2*,2*2) ;

in 3

[vertex] (3) at (2*,1*2) ;

in 2,3

[vertex] (4) at (2*,0*2) ;

[vertex] (21) at (2,4) ;
[vertex] (31) at (2,2) ;
[vertex] (41) at (2,0) ;

[evaluate=as using int(+1)] in 1,2

[edge] (1) – (1);

[evaluate=as using int(+1)] in 1,2

[edge] (4) – (4);

in 1,3

[edge] (1) – (2);

in 1,3

[edge] (2) – (3);
in 1,3

[edge] (3) – (4);

[edge] (42) – (33);
[edge] (12) – (23);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$g_{21}$] at (21) ;
[label=[label distance=-3pt]180:$g_{31}$] at (31) ;

[label=[label distance=-3pt]180:$g_{23}$] at (23) ;
[label=[label distance=-3pt]180:$g_{33}$] at (33) ;

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 3

[vertex] (2) at (2*,2*2) ;

in 3

[vertex] (3) at (2*,1*2) ;

in 2,3

[vertex] (4) at (2*,0*2) ;

[vertex] (21) at (2,4) ;
[vertex] (31) at (2,2) ;
[vertex] (41) at (2,0) ;

[evaluate=as using int(+1)] in 1,2

[edge] (1) – (1);

[evaluate=as using int(+1)] in 1,2

[edge] (4) – (4);

in 1,3

[edge] (1) – (2);

in 1,3

[edge] (2) – (3);
in 1,3

[edge] (3) – (4);

[edge] (41) – (12);
[edge] (42) – (33);
[edge] (12) – (23);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$g_{21}$] at (21) ;
[label=[label distance=-3pt]180:$g_{31}$] at (31) ;

[label=[label distance=-3pt]180:$g_{23}$] at (23) ;
[label=[label distance=-3pt]180:$g_{33}$] at (33) ;

in 2,3

[vertex] (1) at (2*,3*2) ;

in 3

[vertex] (2) at (2*,2*2) ;

in 3

[vertex] (3) at (2*,1*2) ;

in 2,3

[vertex] (4) at (2*,0*2) ;

[vertex] (21) at (2,4) ;
[vertex] (31) at (2,2) ;
[vertex] (41) at (2,0) ;

[evaluate=as using int(+1)] in 2

[edge] (1) – (1);

[evaluate=as using int(+1)] in

[edge] (2) – (2);

[evaluate=as using int(+1)] in

[edge] (3) – (3);

[evaluate=as using int(+1)] in 1,2

[edge] (4) – (4);

in 3

[edge] (1) – (2);

in 1,3

[edge] (2) – (3);
in 3

[edge] (3) – (4);

[edge] (41) – (12);
[edge] (42) – (33);
[edge] (12) – (23);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$g_{21}$] at (21) ;
[label=[label distance=-3pt]180:$g_{31}$] at (31) ;

[label=[label distance=-3pt]180:$g_{23}$] at (23) ;
[label=[label distance=-3pt]180:$g_{33}$] at (33) ;

in 2,3

[vertex] (1) at (2*,3*2) ;

in 3

[vertex] (2) at (2*,2*2) ;

in 3

[vertex] (3) at (2*,1*2) ;

in 2,3

[vertex] (4) at (2*,0*2) ;

[vertex] (21) at (2,4) ;
[vertex] (31) at (2,2) ;
[vertex] (41) at (2,0) ;

[evaluate=as using int(+1)] in 2

[edge] (1) – (1);

[evaluate=as using int(+1)] in

[edge] (2) – (2);

[evaluate=as using int(+1)] in

[edge] (3) – (3);

[evaluate=as using int(+1)] in 1

[edge] (4) – (4);

in 3
[edge] (1) – (2);
in 1,3

[edge] (2) – (3);
in 3

[edge] (3) – (4);

[edge] (41) – (12);
[edge] (12) – (23);
[edge] (13) to[bend right=30] (33);
[edge] (13) to[bend right=30] (43);
[edge] (23) to[bend right=30] (43);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$g_{21}$] at (21) ;
[label=[label distance=-3pt]180:$g_{31}$] at (31) ;

in 3,4,5

[vertex] (1) at (2*,3*2) ;

in 3,4,5

[vertex] (2) at (2*,2*2) ;

in 3,4,5

[vertex] (3) at (2*,1*2) ;

in 3,4,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using int(+1)] in 3,4

[edge] (1) – (1);
[edge] (2) – (2);
[edge] (3) – (3);
[edge] (4) – (4);

in 3,4,5
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[edge] (13) to [bend right=45] (33);
[edge] (13) to [bend right=45] (43);
[edge] (23) to [bend right=45] (43);

[evaluate==int(1+)] in 4

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 5

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

in 3,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=-8pt]45:$g_{23}$] at (23) ;
[label=[label distance=-8pt]-45:$g_{33}$] at (33) ;

in 3,4,5,6,7

[vertex] (1) at (2*,3*2) ;

in 4,5,6,7

[vertex] (2) at (2*,2*2) ;

in 3,4,5,6,7

[vertex] (3) at (2*,1*2) ;

in 3,4,5,6,7

[vertex] (4) at (2*,0*2) ;

[evaluate=as using int(+1)] in 3,4,5,6

[edge] (1) – (1);
[edge] (3) – (3);
[edge] (4) – (4);

[evaluate=as using int(+1)] in 4,5,6

[edge] (2) – (2);

in 3,4,5,6,7

[edge] (3) – (4);

in 4,5,6,7
[edge] (2) – (3);
[edge] (1) – (2);

[edge] (13) to [bend right=45] (33);
[edge] (13) to [bend right=45] (43);

[evaluate==int(1+)] in 4,6

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 5,7

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 5

[edge] (2) – (3);

[evaluate==int(-1)] in 4,6

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (16,5) – (14,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (16,3) – (14,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (16,1) – (14,1);

in 3,...,5,6,7

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=0pt]90:$g_{33}$] at (33) ;
[label=[label distance=-8pt]-30:$g_{24}$] at (24) ;
[label=[label distance=-8pt]30:$g_{34}$] at (34) ;

in 6,7

[vertex] (1) at (2*,3*2) ;

in 6,7

[vertex] (2) at (2*,2*2) ;

in 3,4,5,6,7

[vertex] (3) at (2*,1*2) ;

in 3,4,5,6,7

[vertex] (4) at (2*,0*2) ;

[evaluate=as using int(+1)] in 6

[edge] (1) – (1);

[evaluate=as using int(+1)] in 3,4,5,6

[edge] (3) – (3);

[evaluate=as using int(+1)] in 3,4,5,6

[edge] (4) – (4);

[evaluate=as using int(+1)] in 6

[edge] (2) – (2);

in 3,4,5,6,7

[edge] (3) – (4);

in 6,7
[edge] (2) – (3);

in 6,7

[edge] (1) – (2);

[evaluate==int(1+)] in 6

[edge] (1) – (2);
[evaluate==int(1+)] in 4,6

[edge] (3) – (4);

[evaluate==int(-1)] in 7

[edge] (1) – (2);
[evaluate==int(-1)] in 5,7

[edge] (3) – (4);

[evaluate==int(1+)] in

[edge] (2) – (3);

[evaluate==int(-1)] in 6

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (16,5) – (14,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (16,3) – (14,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (16,1) – (14,1);

in 6,7

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 3,...,5,6,7

[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]90:$g_{33}$] at (33) ;
[label=[label distance=-3pt]90:$g_{34}$] at (34) ;
[label=[label distance=-3pt]90:$g_{35}$] at (35) ;
[label=[label distance=-3pt]180:$g_{26}$] at (26) ;

in 6,7

[vertex] (1) at (2*,3*2) ;

in 6,7

[vertex] (2) at (2*,2*2) ;

in 3,4,5,6,7

[vertex] (3) at (2*,1*2) ;

in 3,4,5,6,7

[vertex] (4) at (2*,0*2) ;

[evaluate=as using int(+1)] in 6

[edge] (1) – (1);

[evaluate=as using int(+1)] in 3,5,6

[edge] (3) – (3);

[evaluate=as using int(+1)] in 3,5,6

[edge] (4) – (4);

[evaluate=as using int(+1)] in 6

[edge] (2) – (2);

in 3,4,5,6,7

[edge] (3) – (4);

in 6,7
[edge] (2) – (3);

in 6,7

[edge] (1) – (2);

[evaluate==int(1+)] in 6

[edge] (1) – (2);
[evaluate==int(1+)] in 6

[edge] (3) – (4);

[evaluate==int(-1)] in 7

[edge] (1) – (2);
[evaluate==int(-1)] in 7

[edge] (3) – (4);

[evaluate==int(1+)] in

[edge] (2) – (3);

[evaluate==int(-1)] in 6

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (16,5) – (14,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (16,3) – (14,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (16,1) – (14,1);

in 6,7

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 3,...,5,6,7

[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=0pt]90:$g_{33}$] at (33) ;
[label=[label distance=0pt]90:$g_{34}$] at (34) ;
[label=[label distance=-3pt]180:$g_{26}$] at (26) ;
[label=[label distance=-5pt]90:$g_{35}$] at (35) ;

Since $N_{}(g_{11}) \subseteq \on{}{g_{22}}$ and $N_{}(g_{41}) \subseteq \on{}{g_{32}}$ in $\Gamma_{n,5}$ and $\Gamma_{n,5} \setminus      g_{22}$, respectively (see <Ref>), using <Ref>, we get that $\ind(\Gamma_{n,5}) \simeq \ind(\Gamma_{n,5} \setminus \{g_{22},g_{32}\})$. Let $\Gamma_{n,5}^1 \coloneq \Gamma_{n,5} \setminus \{g_{22},g_{32}\}$.
Since  $[g_{41},g_{12};g_{21}]$ forms an edge-invariant triplet in $\Gamma_{n,5}^1$ (see <Ref>), from <Ref>, we have $\ind(\Gamma_{n,5}^1) \simeq \ind(\Gamma_{n,5}^1{+}\{g_{41}g_{12}\})$. Let $ \Gamma_{n,5}^2 \eqcolon \Gamma_{n,5}^1{+}\{g_{41}g_{12}\}$.
Similarly, $[g_{31},g_{41};g_{11}]$ forms an edge-invariant triplet in $\Gamma_{n,5}^2$ (see <Ref>), which allows us to delete edge $g_{31}g_{41}$ from $\Gamma_{n,5}^2$ without altering the homotopy type of $\ind(\Gamma_{n,5}^2)$. Now, since $N(g_{31}) \subseteq N(g_{11})$ in $\Gamma_{n,5}^2 - \{g_{31}g_{41}\}$,  using <Ref>, we get that $\ind(\Gamma_{n, 5}^2-\{g_{31}g_{41}\}) \simeq \ind((\Gamma_{n, 5}^2 - \{g_{31}g_{41}\}) \setminus g_{11})$. Let
$\Gamma_{n,5}^3 \eqcolon (\Gamma_{n, 5}^2 - \{g_{31}g_{41}\}) \setminus g_{11}$ (see <Ref>).
Observe that $[g_{13},g_{33};g_{41}]$  is an edge-invariant triplet in
$\Gamma_{n, 5}^3$, $[g_{13},g_{43};g_{41}]$  is an edge-invariant triplet in
$\Gamma_{n, 5}^3 + \{g_{13}g_{33}\}$, and  $[g_{23},g_{43};g_{41}]$  is an edge-invariant triplet in
$\Gamma_{n, 5}^3 + \{g_{13} g_{33}, g_{13}g_{43} \}$. Hence $\ind(\Gamma_{n, 5}^3) \simeq \ind(\Gamma_{n, 5}^3 + \{g_{13} g_{33}, g_{13}g_{43}, g_{23}g_{43}\})$. Furthermore, $[g_{42},g_{33};g_{12}]$ is an edge-invariant triplet in $\Gamma_{n, 5}^3 + \{g_{13} g_{33}, g_{13}g_{43}, g_{23}g_{43} \}$, and $[g_{42},g_{43};g_{12}]$ is an edge-invariant triplet in $\Gamma_{n, 5}^3 + \{g_{13} g_{33}, g_{13}g_{43}, g_{23}g_{43} \}-\{g_{33}g_{42}\}$. Let $\Gamma_{n,5}^4 \coloneq \Gamma_{n, 5}^3 + \{g_{13} g_{33}, g_{13}g_{43}, g_{23}g_{43}\}-\{g_{33}g_{42},g_{42}g_{43}\}$ (see <Ref>).    Then, we have $\ind(\Gamma_{n,5}^3) \simeq \ind(\Gamma_{n,5}^4)$.
In $\Gamma_{n,5}^4$, since $N(g_{42}) \subseteq N(g_{12})$, we have $\ind(\Gamma_{n,5}^4) \simeq \ind(\Gamma_{n,5}^4 \setminus g_{12})$. The  graph, $\Gamma_{n,5}^4 \setminus g_{12}$, consists of three connected components, namely $g_{21}g_{31}, g_{41}g_{42}$ and the third component denoted by $\Gamma_{n,5}^5$ depicted in <Ref>. From <Ref>, it follows that

\begin{equation*}
\ind(\Gamma_{n,5}) \simeq \Sigma^{2}\ind(\Gamma_{n,5}^5).
\end{equation*}

In particular, for $\Gamma_{1,5}$, the graph $\Gamma^5_{1,5}$ is isomorphic to the complete graph $K_4$. Therefore, $\ind(\Gamma_{1,5})\simeq \Sigma^{2}\ind(K_4)\simeq \mathbb{S}^2\vee \mathbb{S}^2 \vee \mathbb{S}^2$.

Now, let $n \geq 2$. Note that, $\Gamma_{n,5}^5\setminus\cn{}{g_{23}}$ is isomorphic to $A_{n-2,5}$ (see <Ref>) and $\ind(\Gamma_{n,5}^5\setminus\cn{}{g_{23}})* \{g_{33}\}$ is a subcomplex of $\ind(\Gamma_{n,5}^5\setminus g_{23})$, which implies that $\ind(\Gamma_{n,5}^5\setminus\cn{}{g_{23}})$ is contractible in $\ind(\Gamma_{n,5}^5\setminus g_{23})$, that is, $\ind(\Gamma_{n,5}^5\setminus\cn{}{g_{23}}) \hookrightarrow \ind(\Gamma_{n,5}^5\setminus g_{23})$ is null homotopic.
Hence from <Ref>, we have

\begin{equation}{\label{eq: Gn lk del}}
\ind(\Gamma_{n,5}) \simeq \Sigma^2\left(\ind(\Gamma_{n,5}^5\setminus g_{23}) \vee \Sigma\ind(A_{n-2,5}) \right).
\end{equation}

We now compute the homotopy type of $\ind(\Gamma_{n,5}^5\setminus g_{23})$.  Let $\Gamma_{n,5}^6 \coloneq \Gamma_{n,5}^5\setminus g_{23}$ and $\Gamma_{n,5}^7 \coloneq \Gamma_{n,5}^6 \setminus N[g_{14}]$ depicted in <Ref>, respectively. We first show that
$\ind(\Gamma_{n,5}^7)$ is contractible in $\ind(\Gamma_{n,5}^6 \setminus g_{14})$.

Since $[g_{34},g_{45};g_{43}]$ and $[g_{35},g_{44};g_{33}]$ are edge-invariant triplets in $\Gamma_{n,5}^7$ and $\Gamma_{n,5}^7 - \{g_{34}g_{45}\}$, respectively, using <Ref>, we have $\ind(\Gamma_{n,5}^7) \simeq \ind(\Gamma_{n,5}^7-\{g_{35}g_{44}$, $g_{34}g_{45}\})$. Similarly, $[g_{34},g_{35}; g_{43}]$ and $[g_{44},g_{45};g_{33}]$ form edge-invariant triplets in $\Gamma_{n,5}^7 - \{g_{34}g_{45}, g_{35}g_{44}\}$ and $(\Gamma_{n,5}^7 - \{g_{34}g_{45},g_{35}g_{44}\})-\{g_{34}g_{35}\}$, respectively.  Let $\Gamma_{n,5}^8 \coloneq (\Gamma_{n,5}^7 - \{g_{34}g_{45},g_{35}g_{44}\}) - \{g_{34}g_{35},g_{44}g_{45}\}$ (see <Ref>). Consequently, we have $\ind(\Gamma_{n,5}^7) \simeq \ind(\Gamma_{n,5}^8) $. Observe that the graph $\Gamma_{n,5}^8$ contains a connected component that is isomorphic to a $4$-cycle induced by vertices $\{g_{33},g_{34},g_{43},g_{44}\}$, and another connected component isomorphic to $B_{n-3,5}$ depicted in <Ref>. Since $N(g_{44})\subseteq N(g_{33})$ and $N(g_{34}) \subseteq N(g_{43})$, we delete vertices $g_{33}$ and $g_{43}$ in $\Gamma_{n,5}^8$ using <Ref>, and have $\ind(\Gamma_{n,5}^8) \simeq \Sigma\ind( B_{n-3,5})$. Furthermore, we have the following sequence

$$\ind(\Gamma_{n,5}^7) \xlongrightarrow{\simeq} \ind(\Gamma_{n,5}^8 \setminus \{g_{33},g_{43}\}) \xhookrightarrow{\text{  Id  }} \ind((\Gamma_{n,5}^{8} \setminus \{g_{33},g_{43}\}) \sqcup \{g_{13}\}) \xhookrightarrow{\text{  Id  }} \ind(\Gamma_{n,5}^6 \setminus g_{14}),

where $\ind((\Gamma_{n,5}^{8} \setminus \{g_{33},g_{43}\}) \sqcup \{g_{13}\}) \simeq \ind(\Gamma_{n,5}^{8} \setminus \{g_{33},g_{43}\})* g_{13}$, and hence contractible. This implies that $\ind(\Gamma_{n,5}^7) \hookrightarrow \ind(\Gamma_{n,5}^6 \setminus g_{14})$ is null homotopic. Therefore, by <Ref>, we have

\begin{equation}{\label{eq: Gamma-n-6}}
\begin{split}
\ind(\Gamma_{n,5}^6) & \simeq \Sigma\ind(\Gamma_{n,5}^7 )\vee \ind(\Gamma_{n,5}^6 \setminus g_{14})
\simeq \Sigma^2 \ind(B_{n-3,5}) \vee \ind(\Gamma_{n,5}^6 \setminus g_{14}),
\end{split}
\end{equation}

where the second equivalence follows from $\ind(\Gamma_{n,5}^7) \simeq \ind(\Gamma_{n,5}^8) \simeq \Sigma \ind(B_{n-3,5})$.
Finally, observe that $g_{13}$ is a simplicial vertex in $\Gamma_{n,5}^6 \setminus g_{14}$, with $N(g_{13})=\{g_{33},g_{43}\}$. Therefore, applying <Ref>, we get

\begin{equation*}
\ind(\Gamma_{n,5}^6 \setminus g_{14}) \simeq \Sigma \ind((\Gamma_{n,5}^6 \setminus g_{14})\setminus N[g_{33}]) \vee \Sigma \ind((\Gamma_{n,5}^6 \setminus g_{14}) \setminus N[g_{43}]).
\end{equation*}

From <Ref> and <Ref>, we observe that $(\Gamma_{n,5}^6 \setminus g_{14})\setminus N[g_{33}] \simeq C_{n-2,5}$ and $(\Gamma_{n,5}^6 \setminus g_{14}) \setminus N[g_{43}] \simeq D_{n-2,5}$. Therefore,

\begin{equation}{\label{eq: G'n-g23 lk del}}
\ind(\Gamma_{n,5}^6 \setminus g_{14}) \simeq \Sigma \ind(C_{n-2,5}) \vee \Sigma \ind(D_{n-2,5}).
\end{equation}

Combining (<ref>), (<ref>), and (<ref>), we get the required recurrence for $n\geq 3$.
However, for $n=2$, we have $\ind(\Gamma_{2,5}) \simeq \Sigma^3\ind(A_{0,5}) \vee \Sigma^4 \ind(B_{-1}) \vee \Sigma^3 \ind(C_{0,5}) \vee \Sigma^3 \ind(D_{0,5}) \simeq \vee_7 \Sp^4$ using <Ref> and the fact that $B_{-1}\cong K_2$.

For $n \geq 1$, $\ind(A_{n,5}) \simeq
\Sigma^2\ind(C_{n-1,5}) \vee \Sigma^2\ind(C_{n-1,5}) \vee \Sigma^2\ind(D_{n-1,5}).$

in 3,4,5

[vertex] (1) at (2*,3*2) ;

in 2,3,4,5

[vertex] (2) at (2*,2*2) ;

in 1,2,...,5

[vertex] (3) at (2*,1*2) ;

in 1,2,3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 3,4

[edge] (1) – (1);

[evaluate=as using +1] in 2,3,4

[edge] (2) – (2);

[evaluate=as using +1] in 1,2,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 1,2,3,...,4

[edge] (4) – (4);

in 3,4,5
[edge] (1) – (2);
in 2,3,4,5

[edge] (2) – (3);

in 1,2,3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 4

[edge] (1) – (2);

[evaluate==int(1+)] in 2,4

[edge] (3) – (4);

[evaluate==int(-1)] in 3,5

[edge] (1) – (2);

[evaluate==int(-1)] in 3,5

[edge] (3) – (4);

[evaluate==int(1+)] in 3

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[vertex] (a2) at (-2,0)  ;

[edge] (a2) – (31) ;
[edge] (a2) – (41) ;

[label=[label distance=-3pt]180:$a_{2}$] at (a2) ;

in 3,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,...,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$g_{22}$] at (22) ;

in 3

[label=[label distance=-3pt]180:$g_{\x1}$] at (1) ;

$A_{n,5} \setminus N[g_{11}]$

in 2,3,4,5

[vertex] (1) at (2*,3*2) ;

in 3,4,5

[vertex] (2) at (2*,2*2) ;

in 3,...,5

[vertex] (3) at (2*,1*2) ;

in 1,2,3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 2,3,4

[edge] (1) – (1);

[evaluate=as using +1] in 3,4

[edge] (2) – (2);

[evaluate=as using +1] in 3,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 1,2,3,...,4

[edge] (4) – (4);

in 3,4,5
[edge] (1) – (2);
in 3,4,5

[edge] (2) – (3);

in 3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 2,4

[edge] (1) – (2);

[evaluate==int(1+)] in 4

[edge] (3) – (4);

[evaluate==int(-1)] in 5

[edge] (1) – (2);

[evaluate==int(-1)] in 3,5

[edge] (3) – (4);

[evaluate==int(1+)] in 3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[vertex] (a2) at (-2,0)  ;

[edge] (a2) – (41) ;

[label=[label distance=-3pt]180:$a_{2}$] at (a2) ;

in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,...,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]180:$g_{23}$] at (23) ;
[label=[label distance=-5pt]180:$g_{33}$] at (33) ;

$A_{n,5} \setminus N[g_{21}]$

Since $a_1$ is a simplicial vertex in $A_{n,5}$ (see <Ref>), with $N(a_1)=\{g_{11},g_{21}\}$, using <Ref>, we have

\begin{equation*}
\ind(A_{n,5}) \simeq \Sigma\ind(A_{n,5} \setminus \n[g_{11}]) \vee \Sigma\ind(A_{n,5} \setminus \n[g_{21}]).
\end{equation*}

Furthermore, $a_2$ is a simplicial vertex in $A_{n,5} \setminus \n[g_{11}]$ with $N(a_2)=\{g_{31},g_{41}\}$, and a simplicial vertex in $A_{n,5} \setminus \n[g_{21}]$ with $N(a_2)=\{g_{41}\}$. Therefore, <Ref> provides us

\begin{align*}
\ind(A_{n,5}) \simeq \Sigma^2\ind(A_{n,5}\setminus\n[g_{11},g_{31}]) &\vee \Sigma^2\ind(A_{n,5}\setminus\n[g_{11},g_{41}])\\
&\vee \Sigma^2\ind(A_{n,5}\setminus\n[g_{21},g_{41}]).
\end{align*}

Note that the graphs $A_{n,5} \setminus \n[g_{11},g_{31}]$ and $A_{n,5} \setminus \n[g_{21},g_{41}]$ are isomorphic to $C_{n-1,5}$, whereas the graph $A_{n,5} \setminus \n[g_{11},g_{41}]$ is isomorphic to $D_{n-1,5}$ (refer to <Ref>). Therefore,
$ \ind(A_{n,5}) \simeq \Sigma^2\ind(C_{n-1,5}) \vee \Sigma^2\ind(D_{n-1,5}) \vee \Sigma^2\ind(C_{n-1,5})$.

For $n \geq 1$, $\ind(B_{n,5}) \simeq \Sigma\ind(C_{n,5}) \vee \Sigma^2\ind(\Lambda_{n-1,5}).$

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,2,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2

[edge] (1) – (1);

[evaluate=as using +1] in 1,2

[edge] (2) – (2);

[evaluate=as using +1] in 1,2

[edge] (3) – (3);

[evaluate=as using +1] in 1,2

[edge] (4) – (4);

in 1,2,3
[edge] (1) – (2);
in 1,2,3

[edge] (2) – (3);

in 1,2,3

[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1

[edge] (2) – (3);

[evaluate==int(-1)] in 2

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[vertex] (b1) at (-2,6)  ;
[vertex] (b2) at (-2,4)  ;
[vertex] (b4) at (-2,0)  ;
[vertex] (b31) at (-4,2)  ;
[vertex] (b41) at (-4,0)  ;

[edge] (b1) – (11) ;
[edge] (b1) – (21) ;
[edge] (b1) – (b2) ;
[edge] (b2) – (21) ;
[edge] (b2) – (11) ;
[edge] (b4) – (41) ;
[edge] (b4) – (31) ;

[edge] (b31) – (b41) ;
[edge] (b41) – (b4) ;
[edge] (b31) – (b2) ;

[edge] (41) – (b2) ;

[label=[label distance=-3pt]180:$b_{3}$] at (b1) ;
[label=[label distance=-3pt]180:$b_{4}$] at (b2) ;
[label=[label distance=-3pt]270:$b_{6}$] at (b4) ;
[label=[label distance=-3pt]180:$b_{1}$] at (b31) ;
[label=[label distance=-3pt]180:$b_{2}$] at (b41) ;

[label=[label distance=-8pt]30:$g_{21}$] at (21) ;
[label=[label distance=-8pt]-30:$g_{31}$] at (31) ;
in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,2,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2

[edge] (1) – (1);

[evaluate=as using +1] in 1,2

[edge] (2) – (2);

[evaluate=as using +1] in 1,2

[edge] (3) – (3);

[evaluate=as using +1] in 1,2

[edge] (4) – (4);

in 1,2,3
[edge] (1) – (2);
in 1,2,3

[edge] (2) – (3);

in 1,2,3

[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1

[edge] (2) – (3);

[evaluate==int(-1)] in 2

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[vertex] (b1) at (-2,6)  ;
[vertex] (b2) at (-2,4)  ;
[vertex] (b4) at (-2,0)  ;
[vertex] (b31) at (-4,2)  ;
[vertex] (b41) at (-4,0)  ;

[edge] (b1) – (11) ;
[edge] (b1) – (21) ;
[edge] (b1) – (b2) ;
[edge] (b2) – (21) ;
[edge] (b2) – (11) ;
[edge] (b4) – (41) ;
[edge] (b4) – (31) ;

[edge] (b31) – (b41) ;
[edge] (b41) – (b4) ;

[edge] (41) – (b2) ;
[edge] (31) – (b2) ;

[label=[label distance=-3pt]180:$b_{3}$] at (b1) ;
[label=[label distance=-3pt]180:$b_{4}$] at (b2) ;
[label=[label distance=-3pt]270:$b_{6}$] at (b4) ;
[label=[label distance=-3pt]180:$b_{1}$] at (b31) ;
[label=[label distance=-3pt]180:$b_{2}$] at (b41) ;

[label=[label distance=-8pt]30:$g_{21}$] at (21) ;
[label=[label distance=-8pt]-30:$g_{31}$] at (31) ;
in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,2,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2

[edge] (1) – (1);

[evaluate=as using +1] in 1,2

[edge] (2) – (2);

[evaluate=as using +1] in 1,2

[edge] (3) – (3);

[evaluate=as using +1] in 1,2

[edge] (4) – (4);

in 1,2,3
[edge] (1) – (2);
in 1,2,3

[edge] (2) – (3);

in 1,2,3

[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1

[edge] (2) – (3);

[evaluate==int(-1)] in 2

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[vertex] (b1) at (-2,6)  ;
[vertex] (b2) at (-2,4)  ;
[vertex] (b31) at (-4,2)  ;
[vertex] (b41) at (-4,0)  ;

[edge] (b1) – (11) ;
[edge] (b1) – (21) ;
[edge] (b1) – (b2) ;
[edge] (b2) – (21) ;
[edge] (b2) – (11) ;

[edge] (b31) – (b41) ;

[edge] (41) – (b2) ;
[edge] (31) – (b2) ;

[label=[label distance=-3pt]180:$b_{3}$] at (b1) ;
[label=[label distance=-3pt]180:$b_{4}$] at (b2) ;
[label=[label distance=-3pt]180:$b_{1}$] at (b31) ;
[label=[label distance=-3pt]180:$b_{2}$] at (b41) ;

[label=[label distance=-8pt]30:$g_{21}$] at (21) ;
[label=[label distance=-8pt]-30:$g_{31}$] at (31) ;
in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

$B_{n,5}^3 \sqcup \{b_1b_2\}$

Since $N(b_2) \subseteq N(b_5)$ in the graph $B_{n,5}$ (see <Ref>),  using <Ref>, we get $\ind(B_{n,5}) \simeq \ind(B_{n,5} \setminus b_5)$. Observe that $[b_4,g_{41};b_2]$ is an edge-invariant triplet in $B_{n,5}\setminus b_5$. Therefore, by <Ref>, we have $\ind(B_{n,5}) \simeq \ind(B_{n,5}^1)$, where $B_{n,5}^1 = (B_{n,5}\setminus b_{5}) {+} \{b_4g_{41}\}$ (see <Ref>). Furthermore, $[b_{4},g_{31};b_2]$ is an edge-invariant triplet in $B_{n,5}^1$, and $[b_1,b_4;b_6]$ is an  edge-invariant triplet in  $B_{n,5}^1+ \{b_4g_{31}\}$. Let $B_{n,5}^2 \coloneqq (B_{n,5}^1{+}\{b_4g_{31}\})- \{b_1b_4\}$ (see <Ref>). Then, $\ind(B_{n,5}) \simeq \ind(B_{n,5}^2)$. Since  $N(b_1) \subseteq N(b_6)$ in $B_{n,5}^2$, using  <Ref>, we have  $\ind(B_{n,5}^2) \simeq \ind(B_{n,5}^2 \setminus b_6)$. Note that the graph $B_{n,5}^2 \setminus b_6$ comprises two connected components: an edge $b_1b_2$ and another component that we call $B_{n,5}^3$ (see <Ref>). This implies that $\ind(B_{n,5}) \simeq \ind(B_{n,5}^2)\simeq \Sigma \ind(B_{n,5}^3)$. Observe that $\ind(B_{n,5}^3 \setminus N[b_4)])*\{b_3\}$ is a subcomplex of $\ind(B_{n,5}^3 \setminus b_4)$, which implies that the inclusion map $\ind(B_{n,5}^3 \setminus N[b_4)]) \hookrightarrow \ind(B_{n,5}^3 \setminus b_4)$ is null homotopic. Therefore, by <Ref>, we get

\begin{equation*}
\ind(B_{n,5}^3) \simeq \ind(B_{n,5}^3 \setminus b_4) \vee \Sigma\ind(B_{n,5}^3 \setminus N[b_4)]).
\end{equation*}

Note that, the graph $B_{n,5}^3 \setminus N[b_4]$ is isomorphic to the graph $\Lambda_{n-1,5}$ and graph $B_{n,5}^3 \setminus b_4$ is isomorphic to $C_{n,5}$ (refer to <Ref> and <Ref>). Further, since $\ind(B_{n,5}) \simeq \Sigma \ind(B_{n,5}^3)$, we have
$\ind(B_{n,5}) \simeq \Sigma\ind(C_{n,5}) \vee \Sigma^2\ind(\Lambda_{n-1,5}).$

For $n\geq 1$,

\begin{equation*}
\ind(C_{n,5}) \simeq \begin{cases} \mathbb{S}^2\vee\mathbb{S}^2\vee \mathbb{S}^2
& \text{if $n = 1$,} \\
\Sigma^{3}\ind(A_{n-2,5}) \bigvee \Sigma^2\ind(E_{n-1,5})\bigvee \Sigma^{3}\ind(F_{n-2,5}) & \text{if $n \geq 2$.}
\end{cases}
\end{equation*}

in 1,2,...,5

[vertex] (1) at (2*,3*2) ;

in 1,3,4,5

[vertex] (2) at (2*,2*2) ;

in 2,...,5

[vertex] (3) at (2*,1*2) ;

in 3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2,3,4

[edge] (1) – (1);

[evaluate=as using +1] in 3,4

[edge] (2) – (2);

[evaluate=as using +1] in 2,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 3,...,4

[edge] (4) – (4);

in 1,3,4,5

[edge] (1) – (2);

in 3,4,5

[edge] (2) – (3);

in 3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 2,4

[edge] (1) – (2);

[evaluate==int(1+)] in 2,4

[edge] (3) – (4);

[evaluate==int(-1)] in 5

[edge] (1) – (2);

[evaluate==int(-1)] in 5

[edge] (3) – (4);

[evaluate==int(1+)] in 1,3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

in 1,2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 3,...,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$g_{32}$] at (32) ;
[label=[label distance=-3pt]-90:$g_{21}$] at (21) ;
[label=[label distance=-7pt]210:$g_{23}$] at (23) ;

in 1,2,...,5

[vertex] (1) at (2*,3*2) ;

in 1,3,4,5

[vertex] (2) at (2*,2*2) ;

in 2,...,5

[vertex] (3) at (2*,1*2) ;

in 3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,3,4

[edge] (1) – (1);

[evaluate=as using +1] in 3,4

[edge] (2) – (2);

[evaluate=as using +1] in 2,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 3,...,4

[edge] (4) – (4);

in 1,3,4,5

[edge] (1) – (2);

in 3,4,5

[edge] (2) – (3);

in 3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 4

[edge] (1) – (2);

[evaluate==int(1+)] in 2,4

[edge] (3) – (4);

[evaluate==int(-1)] in 5

[edge] (1) – (2);

[evaluate==int(-1)] in 5

[edge] (3) – (4);

[evaluate==int(1+)] in 1,3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
in 3,...,5

[label=[label distance=-3pt]270:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$g_{32}$] at (32) ;
[label=[label distance=-3pt]-90:$g_{21}$] at (21) ;
[label=[label distance=-7pt]10:$g_{23}$] at (23) ;

[edge] (13) –(32) ;
[edge] (23) –(32) ;

in 1,2,...,5

[vertex] (1) at (2*,3*2) ;

in 3,4,5

[vertex] (2) at (2*,2*2) ;

in 3,...,5

[vertex] (3) at (2*,1*2) ;

in 2,3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2,3,4

[edge] (1) – (1);

[evaluate=as using +1] in 3,4

[edge] (2) – (2);

[evaluate=as using +1] in 3,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 3,...,4

[edge] (4) – (4);

in 3,4,5

[edge] (1) – (2);

in 3,4,5

[edge] (2) – (3);

in 3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 2,4

[edge] (1) – (2);

[evaluate==int(1+)] in 4

[edge] (3) – (4);

[evaluate==int(-1)] in 5

[edge] (1) – (2);

[evaluate==int(-1)] in 5

[edge] (3) – (4);

[evaluate==int(1+)] in 3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[edge] (42) – (33);
[edge] (42) – (43);

in 1,2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 2,3,...,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]180:$g_{23}$] at (23) ;
[label=[label distance=-5pt]180:$g_{33}$] at (33) ;

in 4,5

[vertex] (1) at (2*,3*2) ;

in 4,5

[vertex] (2) at (2*,2*2) ;

in 3,...,5

[vertex] (3) at (2*,1*2) ;

in 2,3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 4

[edge] (1) – (1);

[evaluate=as using +1] in 4

[edge] (2) – (2);

[evaluate=as using +1] in 3,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 2,3,...,4

[edge] (4) – (4);

in 4,5
[edge] (1) – (2);
in 4,5

[edge] (2) – (3);

in 3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 4

[edge] (1) – (2);

[evaluate==int(1+)] in 4

[edge] (3) – (4);

[evaluate==int(-1)] in 5

[edge] (1) – (2);

[evaluate==int(-1)] in 3,5

[edge] (3) – (4);

[evaluate==int(1+)] in

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

(c15) at (15);
(c16) at (16);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

in 4,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 2,3,4,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$g_{33}$] at (33) ;
[label=[label distance=-5pt]180:$g_{24}$] at (24) ;

Since $N(g_{11}) \subseteq N(g_{22})$ in the graph $C_{n,5}$ (see <Ref>),  by <Ref>, we have $\ind(C_{n,5}) \simeq \ind(C_{n,5} \setminus g_{22})$. We denote the graph $C_{n,5} \setminus g_{22}$ by $C_{n,5}^1$. Since $c_1$ is a simplicial vertex and $N(c_1) = \{g_{31}, g_{41}\}$ in $C_{n,5}^1$, using <Ref>, we get that

\begin{equation}{\label{eq: Cn sim-vertex}}
\ind(C_{n,5}^1) \simeq \Sigma \ind(C_{n,5}^1 \setminus \n[g_{31}]) \vee \Sigma \ind(C_{n,5}^1 \setminus \n[g_{41}]).
\end{equation}

Let $C_{n,5}^2 \coloneqq C_{n,5}^1 \setminus N[g_{41}]$ (see <Ref>). Since $[g_{13},g_{32};g_{11}]$ and $[g_{23},g_{32};g_{11}]$ are edge-invariant triplets in $C_{n,5}^2$ and $C_{n,5}^2{+}\{g_{13}g_{32}\}$, respectively, $\ind(C_{n,5}^2)\simeq \ind(C_{n,5}^2{+}\{g_{13}g_{32},$ $g_{23}g_{32}\})$ using <Ref>. Analogously, $[g_{12},g_{13};g_{21}]$ and $[g_{12},g_{23};g_{21}]$ form edge invariant triplets in $C_{n,5}^2+\{g_{13}g_{32},g_{32}g_{23}\}$ and $C_{n,5}^2+\{g_{13}g_{32},g_{32}g_{23}\}-\{g_{12}g_{13}\}$, respectively. Thus, we have $\ind(C_{n,5}^2+\{g_{13}g_{32},g_{32}g_{23}\}) \simeq \ind(C_{n,5}^2+\{g_{13}g_{32},g_{32}g_{23}\}-\{g_{12}g_{13},g_{12}g_{23}\})$. We denote the  graph $C_{n,5}^2+\{g_{13}g_{32},g_{32}g_{23}\}-\{g_{12}g_{13},g_{12}g_{23}\}$ by $C_{n,5}^3$, depicted in <Ref>. Then, $\ind(C_{n,5}^2)\simeq\ind(C_{n,5}^3)$. Furthermore, $N(g_{12}) \subseteq N(g_{21})$ in $C_{n,5}^3$. Therefore,  by <Ref>, we have $\ind(C_{n,5}^3) \simeq \ind(C_{n,5}^3 \setminus g_{21})$. Observe that the graph $C_{n,5}^3 \setminus g_{21}$ consists of two connected components, one of which is an edge $g_{11}g_{12}$ and the other connected component is isomorphic to $E_{n-1,5}$ (see <Ref>). Thus, we have

\begin{equation}{\label{eq: Cn simeq En-1}}
\ind(C_{n,5}^2)\simeq \Sigma \ind(E_{n-1,5}).
\end{equation}

On the other hand, let $C_{n,5}^4 \coloneqq C_{n,5}^1 \setminus N[g_{31}]$ (see <Ref>). Since $g_{11}$ is a pendant vertex in  $C_{n,5}^4$ with a unique neighbour $g_{12}$, by <Ref>, we have $\ind(C_{n,5}^4) \simeq \Sigma \ind(C_{n,5}^4 \setminus N[g_{12}])$. Let  $C_{n,5}^5 \coloneq  C_{n,5}^4 \setminus N[g_{12}] $ (see <Ref>).  Now, for $n=1$, we have $C_{1,5}^5 \simeq K_3$. Therefore, from  <Ref> and <Ref>, we get that

\ind(C_{1,5}) \simeq \Sigma^2\ind(E_{0,5}) \vee \Sigma^2\ind(K_3).

Since $\ind(E_{0,5})\simeq \Sp^0$ (using <Ref>) and $\ind(K_3) \simeq \Sp^0 \vee \Sp^0$, we have $\ind(C_{1,5}) \simeq \vee_3 \Sp^2$.

Let  $n\geq 2$. Since $g_{42}$ is a simplicial vertex and $N(g_{42})=\{g_{33},g_{43}\}$ in $C_{n,5}^5$,  <Ref> provides us

\begin{equation*}
\ind(C_{n,5}^5) \simeq \Sigma \ind(C_{n,5}^5 \setminus \n[g_{33}]) \vee \Sigma\ind(C_{n,5}^5 \setminus \n[g_{43}]).
\end{equation*}

However, the graph $C_{n,5}^5 \setminus N[g_{33}]$ is isomorphic to the graph $A_{n-2,5}$ and the graph $C_{n,5}^5 \setminus N[g_{43}]$ is isomorphic to $F_{n-2,5}$ (refer to <Ref> and <Ref>). Therefore, we have

\begin{equation}{\label{eq:Cn5-sim-vert}}
\ind(C_{n,5}^4) \simeq \Sigma \ind(C_{n,5}^5) \simeq \Sigma^2\ind(A_{n-2,5}) \vee \Sigma^2 \ind(F_{n-2,5}),
\end{equation}

which, along with <Ref> and <Ref> gives the required homotopy equivalence.

For $n\geq 1$,

\begin{equation*}
\ind(D_{n,5}) \simeq \begin{cases}
\mathbb{S}^2\vee \mathbb{S}^2\vee \mathbb{S}^2 & \text{for $n =1$,} \\
\Sigma^3\ind(\Lambda_{n-2,5}) \vee \Sigma^2\ind(E_{n-1,5}) \vee \Sigma^2 \ind(E_{n-1,5}) & \text{for $n \geq 2$.}
\end{cases}
\end{equation*}

Observe that  $[g_{31},g_{12};d_{1}]$ and $[g_{21},g_{42};d_{2}]$ are edge-invariant triplets in $D_{n,5}$ (see <Ref>) and $D_{n,5}{+}\{g_{31}g_{12}\}$, respectively. Hence by  using <Ref>, we have $\ind(D_{n,5})\simeq \ind(D_{n,5}{+}\{g_{31}g_{12},g_{21}g_{42}\})$. Furthermore, $[d_{1},g_{21};g_{41}]$ and $[d_{2},g_{31};g_{11}]$ are edge-invariant triplets in $D_{n,5}{+}\{g_{31}g_{12},g_{21}g_{42}\}$ and $(D_{n,5}{+}\{g_{31}g_{12},g_{21}g_{42}\})- \{d_1g_{21}\}$, respectively. Let $D_{n,5}^1$ be the graph that we obtain after removing the edges $d_1g_{21}$ and $d_2g_{31}$ from $D_{n,5}{+}\{g_{31}g_{12},g_{21}g_{42}\}$ (see <Ref>). Then, it follows from <Ref> that $\ind(D_{n,5})\simeq\ind(D_{n,5}^1)$.

Now, we proceed with $D_{n,5}^1$. Observe that $[g_{21},g_{41};d_{1}]$, $[g_{12},g_{41};d_{1}]$, and $[d_{2},g_{41};g_{11}]$ are edge-invariant triplets in the graphs $D_{n,5}^1$, $D_{n,5}^1+\{g_{21}g_{41}\}$, and $D_{n,5}^1+\{g_{21}g_{41}, g_{12}g_{41}\}$, respectively. Let $D_{n,5}^2\coloneq(D_{n,5}^1+\{g_{21}g_{41}, g_{41}g_{12}\})-\{d_2g_{41}\}$ (see <Ref>). Then <Ref> implies that $\ind(D_{n,5}^1)\simeq \ind(D_{n,5}^2)$. Note that $N(d_2)\subseteq N(g_{11})$ in the resultant graph $D_{n,5}^2$. Therefore, from <Ref>, we have $\ind(D_{n,5})\simeq \ind(D_{n,5}^2\setminus g_{11})$. Since $[g_{12},g_{33};g_{21}]$, and $[g_{12},g_{43};g_{21}]$ are edge-invariant triplets in the graphs $D_{n,5}^2\setminus g_{11}$ and $(D_{n,5}^2\setminus g_{11})+\{g_{12}g_{33}\}$, respectively, we get $\ind(D_{n,5}^2\setminus g_{11})\simeq \ind((D_{n,5}^2\setminus g_{11})+\{g_{12}g_{33},g_{12}g_{43}\})$. Let us take $D_{n,5}^3\coloneq (D_{n,5}^2\setminus g_{11})+\{g_{12}g_{33},g_{12}g_{43}\}$ (see <Ref>). Then, $\ind(D_{n,5}^2)\simeq \ind(D_{n,5}^3)$. Furthermore, $[g_{31},g_{12};g_{42}]$, and $[g_{12},g_{41};g_{32}]$ are edge-invariant triplets in the graphs $D_{n,5}^3$, and $D_{n,5}^3-g_{31}g_{12}$, respectively. Hence   $\ind(D_{n,5}^3)\simeq \ind(D_{n,5}^3-\{g_{31}g_{12},g_{12}g_{41}\})$. Let us denote the graph $D_{n,5}^3-\{g_{31}g_{12},g_{12}g_{41}\}$ by $D_{n,5}^4$ (see <Ref>). Then, $\ind(D_{n,5})\simeq\ind(D_{n,5}^4)$.

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,2,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2

[edge] () – ();

[edge] (12) – (13) ;

in 1,2,3
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1

[edge] (2) – (3);

[evaluate==int(-1)] in 2

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[vertex] (d1) at (-2,4)  ;
[vertex] (d2) at (-2,2)  ;

[edge] (d1) – (11) ;
[edge] (d1) – (d2) ;
[edge] (d2) – (41) ;

Edges after preprocessing

[edge] (31) – (12) ;
[edge] (21) – (42) ;

in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-7pt]180:$g_{21}$] at (21) ;
[label=[label distance=-7pt]180:$g_{31}$] at (31) ;
in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$d_{1}$] at (d1) ;
[label=[label distance=-3pt]180:$d_{2}$] at (d2) ;

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,2,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

in 1,2,3,4

[evaluate=as using +1] in 1,2

[edge] () – ();

[edge] (12) – (13) ;

in 1,2,3
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1

[edge] (2) – (3);

[evaluate==int(-1)] in 2

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[vertex] (d1) at (-2,4)  ;
[vertex] (d2) at (-2,2)  ;

[edge] (d1) – (11) ;
[edge] (d1) – (d2) ;

Edges after preprocessing

[edge] (31) – (12) ;
[edge] (21) – (42) ;
[edge] (21) to[bend right=30] (41) ;
[edge] (12) – (41) ;

in 1,2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

[label=[label distance=-7pt]180:$g_{21}$] at (21) ;
[label=[label distance=-7pt]180:$g_{31}$] at (31) ;
in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]180:$d_{1}$] at (d1) ;
[label=[label distance=-3pt]180:$d_{2}$] at (d2) ;

[label=[label distance=-3pt]0:$g_{33}$] at (33) ;

in 2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,2,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

in 2,3,4

[evaluate=as using +1] in 1,2

[edge] () – ();

[edge] (12) – (13) ;

in 2,3
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[edge] (21) – (31);
[edge] (31) – (41);

[evaluate==int(1+)] in 2

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1

[edge] (2) – (3);

[evaluate==int(-1)] in 2

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[vertex] (d1) at (-2,4)  ;
[vertex] (d2) at (-2,2)  ;

[edge] (d1) – (d2) ;

[edge] (21) to[bend right=45] (41) ;
[edge] (21) – (42) ;
[edge] (12) – (33) ;
[edge] (12) – (43) ;
[edge] (31) – (12) ;
[edge] (12) – (41) ;

in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]90:$g_{21}$] at (21) ;
[label=[label distance=-8pt]45:$g_{32}$] at (32) ;

[label=[label distance=-3pt]180:$d_{1}$] at (d1) ;
[label=[label distance=-3pt]180:$d_{2}$] at (d2) ;
[label=[label distance=-7pt]180:$g_{31}$] at (31) ;

in 2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,2,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

in 2,3,4

[evaluate=as using +1] in 1,2

[edge] () – ();

[edge] (12) – (13) ;

in 2,3
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[edge] (21) – (31);
[edge] (31) – (41);

[evaluate==int(1+)] in 2

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in 1

[edge] (2) – (3);

[evaluate==int(-1)] in 2

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[vertex] (d1) at (-2.5,4)  ;
[vertex] (d2) at (-2.5,2)  ;

[edge] (d1) – (d2) ;

[edge] (21) to[bend right=45] (41) ;
[edge] (21) – (42) ;
[edge] (12) – (33) ;
[edge] (12) – (43) ;

in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]180:$g_{21}$] at (21) ;
[label=[label distance=-8pt]45:$g_{32}$] at (32) ;

[label=[label distance=-5pt]180:$d_{1}$] at (d1) ;
[label=[label distance=-5pt]180:$d_{2}$] at (d2) ;
[label=[label distance=-6pt]94:$g_{22}$] at (22) ;
[label=[label distance=-5pt]360:$g_{33}$] at (33) ;

in 2,3

[vertex] (1) at (2*,3*2) ;

in 2,3

[vertex] (2) at (2*,2*2) ;

in 1,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 2

[edge] (2) – (2);

[evaluate=as using +1] in

[edge] (3) – (3);

[evaluate=as using +1] in 1,2

[edge] (4) – (4);

[edge] (12) – (13) ;

in 2,3

[edge] (1) – (2);

in 3

[edge] (2) – (3);

in 3

[edge] (3) – (4);

[edge] (31) – (41);

[evaluate==int(1+)] in 2

[edge] (1) – (2);

[evaluate==int(-1)] in 3

[edge] (1) – (2);
[edge] (3) – (4);

[evaluate==int(1+)] in

[edge] (2) – (3);

[evaluate==int(-1)] in 2

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[vertex] (d1) at (-2,4)  ;
[vertex] (d2) at (-2,2)  ;

[edge] (d1) – (d2) ;

[edge] (12) – (33) ;
[edge] (12) – (43) ;
[edge] (22) – (33) ;
[edge] (22) – (43) ;

in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-4pt]-180:$g_{22}$] at (22) ;
[label=[label distance=-4pt]0:$g_{23}$] at (23) ;
[label=[label distance=-4pt]0:$g_{33}$] at (33) ;
[label=[label distance=-4pt]90:$g_{31}$] at (31) ;
[label=[label distance=-3pt]180:$d_{1}$] at (d1) ;
[label=[label distance=-3pt]180:$d_{2}$] at (d2) ;

in 2,3,4

[vertex] (1) at (2*,3*2) ;

in 2,3,4

[vertex] (2) at (2*,2*2) ;

in 1,3,4

[vertex] (3) at (2*,1*2) ;

in 1,3,4

[vertex] (4) at (2*,0*2) ;

in 2

[evaluate=as using +1] in 2

[edge] () – ();

[edge] (12) – (13) ;

in 1,2,3,4
[edge] (3) – (4);

[edge] (33) – (24) ;
[edge] (23) – (34) ;
in 3,4
[edge] (1) – (2);
[edge] (2) – (3);
[edge] (3) – (4);

[edge] (12) – (22);
[edge] (31) – (41);

[evaluate==int(1+)] in 2

[edge] (1) – (2);

[evaluate==int(-1)] in 3

[edge] (1) – (2);

[dotted, thick,shorten >=4pt, shorten <=4pt] (8,5) – (10,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (8,3) – (10,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (8,1) – (10,1);

[vertex] (d1) at (-2,4)  ;
[vertex] (d2) at (-2,2)  ;

[edge] (d1) – (d2) ;

[edge] (12) – (33) ;
[edge] (12) – (43) ;

[edge] (22) – (33) ;
[edge] (22) – (43) ;

in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 3

[label=[label distance=-7pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]90:$g_{14}$] at (14) ;
[label=[label distance=-3pt]270:$g_{44}$] at (44) ;
[label=[label distance=-3pt]180:$d_{1}$] at (d1) ;
[label=[label distance=-3pt]180:$d_{2}$] at (d2) ;
[label=[label distance=-3pt]90:$g_{31}$] at (31) ;
[label=[label distance=-5pt]180:$g_{22}$] at (22) ;
[label=[label distance=-3pt]-90:$g_{41}$] at (41) ;

Observe that $(\ind(D_{n,5}^4 \setminus \n[g_{21}])*\{g_{41}\} \subseteq\ind(D_{n,5}^4 \setminus g_{21})$, i.e., $\ind(D_{n,5}^4 \setminus \n[g_{21}])$ is contractible in $\ind(D_{n,5}^4 \setminus g_{21})$. Therefore, <Ref> implies that

\begin{equation}{\label{eq: D'n}}
\begin{split}
\ind(D_{n,5}^4) &\simeq \ind(D_{n,5}^4 \setminus g_{21}) \vee \Sigma\ind(D_{n,5}^4 \setminus \n[g_{21}]) \\
&\simeq \ind(D_{n,5}^4 \setminus g_{21}) \vee \Sigma^2 \ind(E_{n-1,5}),
\end{split}
\end{equation}

where the second homotopy type is coming from the fact that the graph $D_{n,5}^4 \setminus \n[g_{21}]$ has two connected components, one component is an edge $d_1d_2$, and the other component is isomorphic to the graph $E_{n-1,5}$ (see <Ref>).

Now, consider the graph $D_{n,5}^4 \setminus g_{21}$. Since $N(g_{41})\subseteq N(g_{32})$ in $D_{n,5}^4 \setminus g_{21}$, we have  $\ind(D_{n,5}^4 \setminus g_{21}) \simeq \ind(D_{n,5}^4 \setminus \{g_{21}, g_{32}\})$. Furthermore, $[g_{22},g_{33};g_{41}]$, and $[g_{22},g_{43};g_{41}]$ are edge-invariant triplets in the graphs $D_{n,5}^4 \setminus \{g_{21},g_{32}\}$ and $(D_{n,5}^4 \setminus \{g_{21},g_{32}\})+\{g_{22}g_{33}\}$, respectively. Therefore,  we have $\ind( D_{n,5}^4 \setminus g_{21})\simeq \ind((D_{n,5}^4 \setminus \{g_{21},g_{32}\})+\{g_{22}g_{33},g_{22}g_{43}\})$. Let $D_{n,5}^5$ (see <Ref>) denote the graph $(D_{n,5}^4 \setminus \{g_{21},g_{32}\})+\{g_{22}g_{33},g_{22}g_{43}\}$. Then, $\ind(D_{n,5}^4 \setminus g_{21})\simeq \ind(D_{n,5}^5)$.

Since $[g_{22},g_{31};g_{42}]$ is an edge-invariant triplet in the graph $D_{n,5}^5$, we obtain that $\ind(D_{n,5}^5)\simeq \ind(D_{n,5}^5- \{g_{22}g_{31}\})$. Moreover, $N(g_{31})\subseteq N(g_{42})$ in the graph $D_{n,5}^5- \{g_{22}g_{31}\}$ and therefore   $\ind(D_{n,5}^5- \{g_{22}g_{31}\}) \simeq \ind((D_{n,5}^5- \{g_{22}g_{31}\}) \setminus g_{42})$.  Let $D_{n,5}^6$ denote the graph $(D_{n,5}^5- \{g_{22}g_{31}\}) \setminus g_{42}$. Then, $\ind( D_{n,5}^4 \setminus g_{21})\simeq \ind(D_{n,5}^5) \simeq \ind(D_{n,5}^6)$.

Suppose $n=1$. Since $N(g_{13})\subseteq N(g_{33})$ in the graph $D_{1,5}^6$,  we get $\ind(D_{1,5}^6)\simeq \ind(D_{1,5}^6\setminus\{g_{33}\})$. Furthermore, $N(g_{43})\subseteq N(g_{13})\cap N(g_{23})$ in the graph $D_{1,5}^6\setminus\{g_{33}\}$. Therefore, a repeated application of <Ref> provides us with $\ind(D_{1,5}^6)\simeq \ind(D_{1,5}^6\setminus\{g_{33},g_{13},g_{23}\})$. Since the graph $D_{1,5}^6\setminus\{g_{33},g_{13},g_{23}\}$ is isomorphic to the disjoint union of $K_3$ and two edges, we have, $\ind (D_{1,5}^6)\simeq \mathbb{S}^2\vee \mathbb{S}^2$. Hence $\ind(D_{1,5}^4 \setminus g_{21})\simeq \ind(D_{1,5}^6) \simeq \mathbb{S}^2\vee \mathbb{S}^2$. Moreover, from <Ref>, it follows that $\Sigma^2 \ind(E_{0,5})\simeq \mathbb{S}^2$. Hence, $\ind(D_{1,5})\simeq \mathbb{S}^2\vee \mathbb{S}^2\vee \mathbb{S}^2$.

Now, consider $n\geq 2$. Since $\ind(D_{n,5}^6 \setminus \n[g_{12}])*\{g_{22}\} \subseteq \ind(D_{n,5}^6 \setminus g_{12})$, from <Ref>, we have

\begin{equation}{\label{eq: D''n}}
\begin{split}
\ind(D_{n,5}^6) \simeq  \ind(D_{n,5}^6 \setminus g_{12}) \vee \Sigma\ind(D_{n,5}^6 \setminus \n[g_{12}]).
\end{split}
\end{equation}

However, the graph $D_{n,5}^6 \setminus g_{12}$ consists of three connected components, two of the components consist of a single edge, and the third component is isomorphic to $E_{n-1,5}$. Similarly, the graph $D_{n,5}^6 \setminus \n[g_{12}]$ has three connected components, two are distinct edges, and the third component is isomorphic to $\Lambda_{n-2,5}$. Therefore, from <Ref>, we get $\ind(D_{n,5}^6)\simeq \Sigma^2\ind(E_{n-1,5}) \vee \Sigma^3\ind(\Lambda_{n-2,5})$. Hence, from <Ref>, we get that $\ind(D_{n,5}) \simeq \Sigma^2\ind(E_{n-1,5}) \vee \Sigma^3\ind(\Lambda_{n-2,5}) \vee \Sigma^2 \ind(E_{n-1,5})$.

For $n \geq 1$, we have

\begin{align*}
\ind(\Lambda_{n,5}) \simeq
& \left(\bigvee_2 \Sigma^2\ind(A_{n-1,5})\right) \vee  \left(\bigvee_2 \Sigma^2\ind(C_{n-1,5})\right) \\ &\vee  \left(\bigvee_2 \Sigma^2\ind(F_{n-1,5})\right) \vee \Sigma^2\ind(D_{n-1,5}).
\end{align*}

in 2,...,5

[vertex] (1) at (2*,3*2) ;

in 2,3,4,5

[vertex] (2) at (2*,2*2) ;

in 1,2,...,5

[vertex] (3) at (2*,1*2) ;

in 1,2,3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 2,3,4

[edge] (1) – (1);

[evaluate=as using +1] in 2,3,4

[edge] (2) – (2);

[evaluate=as using +1] in 1,2,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 1,2,3,...,4

[edge] (4) – (4);

in 2,3,4,5

[edge] (1) – (2);

in 2,3,4,5

[edge] (2) – (3);

in 1,2,3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 2,4

[edge] (1) – (2);

[evaluate==int(1+)] in 2,4

[edge] (3) – (4);

[evaluate==int(-1)] in 3,5

[edge] (1) – (2);

[evaluate==int(-1)] in 3,5

[edge] (3) – (4);

[evaluate==int(1+)] in 3

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-3pt]-90:$g_{41}$] at (41) ;
[label=[label distance=-5pt]180:$g_{31}$] at (31) ;
[label=[label distance=-5pt]180:$g_{22}$] at (22) ;

[vertex] (l4) at (-2,0)  ;

[edge] (l4) – (41) ;
[edge] (l4) – (31) ;

[label=[label distance=-3pt]180:$l_{4}$] at (l4) ;

$\Lambda_{n,5}\setminus N[l_2]$

in 3,...,5

[vertex] (1) at (2*,3*2) ;

in 2,3,4,5

[vertex] (2) at (2*,2*2) ;

in 1,2,...,5

[vertex] (3) at (2*,1*2) ;

in 1,2,3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 3,4

[edge] (1) – (1);

[evaluate=as using +1] in 2,3,4

[edge] (2) – (2);

[evaluate=as using +1] in 1,2,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 1,2,3,...,4

[edge] (4) – (4);

in 3,4,5

[edge] (1) – (2);

in 2,3,4,5

[edge] (2) – (3);

in 1,2,3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 4

[edge] (1) – (2);

[evaluate==int(1+)] in 2,4

[edge] (3) – (4);

[evaluate==int(-1)] in 3,5

[edge] (1) – (2);

[evaluate==int(-1)] in 3,5

[edge] (3) – (4);

[evaluate==int(1+)] in 3

[edge] (2) – (3);

[evaluate==int(-1)] in 2,4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

in 3,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,...,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]180:$g_{22}$] at (22) ;
[label=[label distance=-3pt]90:$g_{31}$] at (31) ;

[vertex] (l3) at (-2,2)  ;
[vertex] (l4) at (-2,0)  ;

[edge] (l3) – (l4) ;

[edge] (l3) – (31) ;
[edge] (l4) – (41) ;

[edge] (l4) – (31) ;
[edge] (l3) – (41) ;

[label=[label distance=-3pt]180:$l_{3}$] at (l3) ;
[label=[label distance=-3pt]180:$l_{4}$] at (l4) ;

in 2,...,5

[vertex] (1) at (2*,3*2) ;

in 3,4,5

[vertex] (2) at (2*,2*2) ;

in 3,...,5

[vertex] (3) at (2*,1*2) ;

in 1,2,3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 2,3,4

[edge] (1) – (1);

[evaluate=as using +1] in 3,4

[edge] (2) – (2);

[evaluate=as using +1] in 3,4

[edge] (3) – (3);

[evaluate=as using +1] in 1,2,3,...,4

[edge] (4) – (4);

in 3,4,5

[edge] (1) – (2);

in 3,4,5

[edge] (2) – (3);

in 3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 2,4

[edge] (1) – (2);

[evaluate==int(1+)] in 4

[edge] (3) – (4);

[evaluate==int(-1)] in 5

[edge] (1) – (2);

[evaluate==int(-1)] in 3,5

[edge] (3) – (4);

[evaluate==int(1+)] in 3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,...,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]180:$g_{23}$] at (23) ;
[label=[label distance=-5pt]180:$g_{33}$] at (33) ;

[vertex] (l3) at (-2,2)  ;
[vertex] (l4) at (-2,0)  ;

[edge] (l3) – (l4) ;
[edge] (l4) – (41) ;
[edge] (l3) – (41) ;

[label=[label distance=-3pt]180:$l_{3}$] at (l3) ;
[label=[label distance=-3pt]180:$l_{4}$] at (l4) ;

The proof follows primarily from the application of <Ref>. Since $l_1$ is a simplicial vertex and $N(l_1)=\{g_{11},g_{21},l_2\}$ in $\Lambda_{n,5}$, we have

\begin{equation}{\label{eq: En-e1-sim}}
\ind(\Lambda_{n,5}) \simeq \Sigma \ind(\Lambda_{n,5} \setminus \n[g_{11}]) \vee \Sigma \ind(\Lambda_{n,5} \setminus \n[g_{21}]) \vee \Sigma \ind(\Lambda_{n,5} \setminus \n[l_{2}]).
\end{equation}

Note that $\Lambda_{n,5} \setminus \n[l_{2}]$ is isomorphic to $C_{n+1}^5$ (see <Ref> in <Ref>). Therefore, it follows from <Ref> that

\begin{equation}{\label{eq: En-e2}}
\ind(\Lambda_{n,5} \setminus \n[l_{2}]) \simeq \Sigma \ind(A_{n-1,5}) \vee \Sigma\ind(F_{n-1,5}).
\end{equation}

Let $\Lambda_{n,5}^1 := \Lambda_{n,5} \setminus \n[g_{11}]$ (see <Ref>) and  $\Lambda_{n,5}^2 := \Lambda_{n,5} \setminus \n[g_{21}]$ (see <Ref>). Since $l_4$ is a simplicial vetyex in $\Lambda_{n,5}^1$, with $N(l_4)=\{ l_3,g_{31},g_{41}\}$, using <Ref>, we have,

\begin{equation*}
\begin{split}
\ind(\Lambda_{n,5}^1) &\simeq \Sigma \ind(\Lambda_{n,5}^1 \setminus \n[l_3]) \vee \Sigma \ind(\Lambda_{n,5}^1 \setminus \n[g_{31}]) \vee \Sigma \ind(\Lambda_{n,5}^1 \setminus \n[g_{41}]).
\end{split}
\end{equation*}

Observe that the graphs $\Lambda_{n,5}^1 \setminus \n[l_3]$, $\Lambda_{n,5}^1 \setminus \n[g_{31}]$, and $\Lambda_{n,5}^1 \setminus \n[g_{41}]$ are isomorpphic to the graphs $F_{n-1,5}$,  $C_{n-1,5}$ and $D_{n-1,5}$, respectively (see  <Ref>, <ref> and <ref> respectively). Thus, we have
\begin{equation}{\label{En-e21-sim-e4}}
\begin{split}
\ind(\Lambda_{n,5}^1) &\simeq \Sigma \ind(F_{n-1,5}) \vee \Sigma \ind(C_{n-1,5}) \vee \Sigma \ind(D_{n-1,5}).
\end{split}
\end{equation}

Further, for the simplicial vertex $l_3$ with $N(l_3)=\{l_4,g_{41}\}$ in $\Lambda_{n,5}^2$,  we have

\begin{equation}{\label{En-e22-sim-e3}}
\begin{split}
\ind(\Lambda_{n,5}^2) &\simeq \Sigma \ind(\Lambda_{n,5}^2 \setminus \n[l_4]) \vee \Sigma \ind(\Lambda_{n,5}^2 \setminus \n[g_{41}]) \\
&\simeq \Sigma \ind(A_{n-1,5}) \vee \Sigma \ind(C_{n-1,5}),
\end{split}
\end{equation}

wherein the graphs $\Lambda_{n,5}^2 \setminus \n[l_4]$ and $\Lambda_{n,5}^2 \setminus \n[g_{41}]$ are isomorphic to graphs $A_{n-1,5}$ and $C_{n-1,5}$, respectively.
Combining (<ref>), (<ref>), (<ref>), and (<ref>), we get the desired result.

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,2,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2

[edge] (1) – (1);

[evaluate=as using +1] in 1,2

[edge] (2) – (2);

[evaluate=as using +1] in 1,2

[edge] (3) – (3);

[evaluate=as using +1] in 1,2

[edge] (4) – (4);

in 1,2,3

[edge] (1) – (2);

in 1,2,3

[edge] (2) – (3);

in 1,2,3

[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);

[evaluate==int(1+)] in 2

[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);

[evaluate==int(-1)] in 3

[edge] (3) – (4);

[evaluate==int(1+)] in

[edge] (2) – (3);

[evaluate==int(-1)] in

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-7pt]10:$g_{32}$] at (32) ;
[label=[label distance=-7pt]10:$g_{21}$] at (21) ;
[label=[label distance=-7pt]10:$g_{31}$] at (31) ;

[vertex] (e1) at (-1.5,3)  ;

[edge] (e1) – (11) ;
[edge] (e1) – (21) ;
[edge] (e1) – (31) ;
[edge] (e1) – (41) ;

[label=[label distance=-3pt]180:$e_{1}$] at (e1) ;

in 1,2,3,4

[vertex] (1) at (2*,3*2) ;

in 1,2,3,4

[vertex] (2) at (2*,2*2) ;

in 1,2,3,4

[vertex] (3) at (2*,1*2) ;

in 1,2,3,4

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2,3

[edge] (1) – (1);

[evaluate=as using +1] in 1,2,3

[edge] (2) – (2);

[evaluate=as using +1] in 1,2,3

[edge] (3) – (3);

[evaluate=as using +1] in 1,2,3

[edge] (4) – (4);

in 1,2,3,4

[edge] (1) – (2);

in 1,2,3,4

[edge] (2) – (3);

in 1,2,3,4

[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);

[evaluate==int(1+)] in 2

[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);

[evaluate==int(-1)] in 3

[edge] (3) – (4);

[edge] (23) – (34);
[edge] (33) – (24);

[edge] (11) – (32);
[edge] (41) – (32);

[dotted, thick,shorten >=4pt, shorten <=4pt] (8,5) – (10,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (8,1) – (10,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-7pt]10:$g_{32}$] at (32) ;
[label=[label distance=-7pt]30:$g_{21}$] at (21) ;

[vertex] (e1) at (-1.5,3)  ;

[edge] (e1) – (11) ;
[edge] (e1) – (21) ;
[edge] (e1) – (31) ;
[edge] (e1) – (41) ;

[label=[label distance=-3pt]180:$e_{1}$] at (e1) ;

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2

[edge] (1) – (1);

[evaluate=as using +1] in 1,2

[edge] (2) – (2);

[evaluate=as using +1] in

[edge] (3) – (3);

[evaluate=as using +1] in 1,2

[edge] (4) – (4);

in 1,2,3

[edge] (1) – (2);

in 1,3

[edge] (2) – (3);

in 1,3

[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);

[evaluate==int(1+)] in

[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);

[evaluate==int(-1)] in 3

[edge] (3) – (4);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-7pt]30:$g_{21}$] at (21) ;
[label=[label distance=-7pt]-10:$g_{22}$] at (22) ;
[label=[label distance=-7pt]30:$g_{31}$] at (31) ;

[vertex] (e1) at (-1.5,3)  ;

[edge] (e1) – (11) ;
[edge] (e1) – (21) ;
[edge] (e1) – (31) ;
[edge] (e1) – (41) ;

[label=[label distance=-3pt]180:$e_{1}$] at (e1) ;

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,2,3

[vertex] (2) at (2*,2*2) ;

in 1,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2

[edge] (1) – (1);

[evaluate=as using +1] in 1,2

[edge] (2) – (2);

[evaluate=as using +1] in

[edge] (3) – (3);

[evaluate=as using +1] in 1,2

[edge] (4) – (4);

in 1,2,3

[edge] (1) – (2);

in 1,3

[edge] (2) – (3);

in 1,3

[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);

[evaluate==int(1+)] in

[edge] (3) – (4);

[evaluate==int(-1)] in 3

[edge] (1) – (2);

[evaluate==int(-1)] in 3

[edge] (3) – (4);

[edge] (22) – (11);
[edge] (22) – (41);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-8pt]-10:$g_{22}$] at (22) ;
[label=[label distance=-8pt]-30:$g_{21}$] at (21) ;

[vertex] (e1) at (-1.5,3)  ;

[edge] (e1) – (11) ;
[edge] (e1) – (21) ;
[edge] (e1) – (31) ;
[edge] (e1) – (41) ;

[label=[label distance=-3pt]180:$e_{1}$] at (e1) ;

in 1,2,3

[vertex] (1) at (2*,3*2) ;

in 1,3

[vertex] (2) at (2*,2*2) ;

in 1,3

[vertex] (3) at (2*,1*2) ;

in 1,2,3

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2

[edge] (1) – (1);

[evaluate=as using +1] in

[edge] (2) – (2);

[evaluate=as using +1] in

[edge] (3) – (3);

[evaluate=as using +1] in 1,2

[edge] (4) – (4);

in 1,3

[edge] (1) – (2);

in 1,3

[edge] (2) – (3);

in 1,3

[edge] (3) – (4);

[evaluate==int(1+)] in 2

[edge] (1) – (2);

[evaluate==int(1+)] in

[edge] (3) – (4);

[evaluate==int(-1)] in

[edge] (1) – (2);

[evaluate==int(-1)] in 3

[edge] (3) – (4);

[dotted, thick,shorten >=4pt, shorten <=4pt] (6,5) – (8,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,3) – (8,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (6,1) – (8,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,3

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 1,2,3

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]0:$g_{21}$] at (21) ;
[label=[label distance=-5pt]0:$g_{31}$] at (31) ;

[vertex] (e1) at (-1.5,3)  ;

[edge] (e1) – (11) ;
[edge] (e1) – (21) ;
[edge] (e1) – (31) ;
[edge] (e1) – (41) ;

[label=[label distance=-3pt]180:$e_{1}$] at (e1) ;

For $n\geq 1$,

\begin{equation*}
\small
\ind(E_{n,5}) \simeq \begin{cases}
\bigvee_{10} \mathbb{S}^2 & \text{for $n = 1$,} \\
\Sigma^3\left((\vee_2 \ind(A_{n-2,5})) \bigvee (\vee_2 \ind(F_{n-2,5}))\right) \bigvee \Sigma\ind(A_{n-1,5}) \bigvee \ind(\Gamma_{n,5})
& \text{for $n \geq 2$.}
\end{cases}
\end{equation*}

Observe that $[g_{21},g_{32};g_{41}]$ and $[g_{22},g_{31};g_{11}]$ are edge-invariant triplets in the graphs $E_{n,5}$ and $E_{n,5} - \{g_{21}g_{32}\}$, respectively. Thus, we have $\ind(E_{n,5}) \simeq \ind(E_{n,5} - \{g_{21}g_{32},g_{22}g_{31}\})$. Let $E_{n,5}^1 \coloneq E_{n,5} - \{g_{21}g_{32},g_{22}g_{31}\}$ (see <Ref>). Since $[g_{11},g_{32};g_{41}]$ and $[g_{41},g_{32};g_{21}]$ are edge-invariant triplets in $E_{n,5}^1$ and $E_{n,5}^1{+}\{g_{11}g_{32}\}$, respectively,  we have $\ind(E_{n,5}^1) \simeq \ind(E_{n,5}^1+\{g_{11}g_{32},g_{41}g_{32}\})$. Let $E_{n,5}^2 \coloneq E_{n,5}^1{+}\{g_{11}g_{32},g_{41}g_{32}\}$ (see <Ref>). Note that $\ind(E_{n,5}^2 \setminus N[g_{32}])*\{g_{42}\}$ is a subcomplex of $\ind(E_{n,5}^2\setminus g_{32})$. This implies that $\ind(E_{n,5}^2 \setminus N[g_{32}]) \hookrightarrow \ind(E_{n,5}^2\setminus g_{32})$ is null homotopic, and by <Ref>, we have

\begin{equation}{\label{En first}}
\ind(E_{n,5})\simeq \ind(E_{n,5}^2) \simeq \ind(E_{n,5}^2\setminus g_{32}) \vee \Sigma \ind(E_{n,5}^2 \setminus N[g_{32}]).
\end{equation}

The graph $E_{n,5}^2 \setminus N[g_{32}]$ has two connected components; one component consists of the edge $e_1g_{21}$, and the other component is isomorphic to $C_{n,5}^5$ (see <Ref>). Therefore,

\begin{equation}{\label{En2-g32}}
\ind(E_{n,5}^2 \setminus N[g_{32}]) \simeq \Sigma\ind(C_{n,5}^5).
\end{equation}

Let $E_{n,5}^3 \coloneq E_{n,5}^2 \setminus g_{32}$ (see <Ref>). Using <Ref>, for the edge-invariant triplets $[g_{22},g_{41};g_{11}]$ and $[g_{22},g_{11};g_{31}]$ in $E_{n,5}^3$ and $E_{n,5}^3{+}\{g_{22}g_{41}\}$, respectively, we get $\ind(E_{n,5}^3) \simeq \ind(E_{n,5}^3{+}\{g_{22}g_{11},g_{22}g_{41}\})$. Let $E_{n,5}^4 \coloneq E_{n,5}^3{+}\{g_{22}g_{11},g_{22}g_{41}\}$ (see <Ref>). Then $\ind(E_{n,5}^3)\simeq \ind(E_{n,5}^4)$. Furthermore,  $\ind(E_{n,5}^4 \setminus N[g_{22}])*\{g_{12}\}$ is a subcomplex of $\ind(E_{n,5}^4 \setminus g_{22})$. Therefore, using <Ref>, we have

\begin{equation*}
\ind(E_{n,5}^3) \simeq\ind(E_{n,5}^4) \simeq \ind(E_{n,5}^4 \setminus g_{22}) \vee \Sigma \ind(E_{n,5}^4\setminus N[g_{22}]).
\end{equation*}

The graph $E_{n,5}^4\setminus N[g_{22}]$ has two connected components; one component consists of the edge $e_1g_{31}$, and the other component is a graph isomorphic to $C_{n,5}^5$ (see <Ref>). Hence

\begin{equation}{\label{En4-link-del}}
\ind(E_{n,5}^3) \simeq \ind(E_{n,5}^4 \setminus g_{22}) \vee \Sigma^2 \ind(C_{n,5}^5).
\end{equation}

Finally, let $E_{n,5}^5 \coloneq E_{n,5}^4 \setminus g_{22}$. Since $\ind(E_{n,5}^5 \setminus \n[e_1])*\{g_{21}\} \subseteq \ind(E_{n,5}^5 \setminus e_1)$, the inclusion map  $\ind(E_{n,5}^5 \setminus \n[e_1])*\{g_{21}\} \hookrightarrow \ind(E_{n,5}^5 \setminus e_1)$ is null homotopic. Therefore, by <Ref>, we have

\begin{equation*}
\begin{split}
\ind(E^5_{n,5}) &\simeq \ind(E^5_{n,5} \setminus e_{1}) \vee \Sigma\ind(E^5_{n,5} \setminus \n[e_{1}]).
\end{split}
\end{equation*}

Further, observe that the graph $E^5_{n,5} \setminus e_1$ is isomorphic to $\Gamma_{n,5}\setminus\{g_{22},g_{32}\}$ (see <Ref>, and <Ref>), and the graph  $E_{n,5}^5 \setminus N[e_1]$ is isomorphic to $A_{n-1,5}$ (see <Ref>, and <Ref>). Therefore, $\ind(E^5_{n,5} \setminus e_1) \simeq \ind(\Gamma_{n,5}\setminus\{g_{22},g_{32}\})$ and $\ind(E_{n,5}^5 \setminus N[e_1]) \simeq \ind(A_{n-1,5})$. Since $N(g_{11}) \subseteq N(g_{22})$ and $N(g_{41}) \subseteq N(g_{32})$ in $\Gamma_{n, 5}$ and $\Gamma_{n, 5} \setminus g_{22}$, respectively (see <Ref>), we follow that $\ind(\Gamma_{n, 5}) \simeq \ind(\Gamma_{n, 5} \setminus \{g_{22}, g_{33}\})$. Therefore,

\begin{equation}{\label{En5-link-del}}
\ind(E^5_{n,5}) \simeq \ind(\Gamma_{n,5}) \vee \Sigma\ind(A_{n-1,5}).
\end{equation}

Combining  <Ref>, we get

\begin{equation}{\label{En3-final}}
\ind(E_{n,5}^3) \simeq \ind(\Gamma_{n,5}) \vee \Sigma\ind(A_{n-1,5}) \vee \Sigma^2 \ind(C_{n,5}^5).
\end{equation}

Thus, combining the homotopy equivalences in <Ref>, we get

\begin{equation}{\label{En pre}}
\ind(E_{n,5}) \simeq \Sigma^2 \left(\vee_2 \ind(C_{n,5}^5)\right) \vee \Sigma\ind(A_{n-1,5}) \vee \ind(\Gamma_{n,5}).
\end{equation}

It follows from <Ref> that $\ind(A_{0,5})\simeq \mathbb{S}^1\vee\mathbb{S}^1\vee\mathbb{S}^1$, $\ind(\Gamma_{1,5})\simeq \mathbb{S}^2\vee\mathbb{S}^2\vee\mathbb{S}^2 $ and $\ind(C_{1,5}^5)\simeq \mathbb{S}^0\vee\mathbb{S}^0$ (because $C_{1,5}^5 \cong K_3$). Therefore, $\ind(E_{1,5})\simeq \bigvee_{10} \mathbb{S}^2.$

For $n\geq 2$, we have $\ind(C_{n,5}^5) \simeq  \Sigma( \ind(A_{n-2,5}) \vee  \ind(F_{n-2,5}))$ from <Ref>. Therefore, from the homotopy equivalence in <Ref>, we have for $n\geq 2$,

\begin{equation*}
\ind(E_{n,5}) \simeq  \Sigma^3\left((\vee_2 \ind(A_{n-2,5})) \bigvee (\vee_2 \ind(F_{n-2,5}))\right) \vee \Sigma\ind(A_{n-1,5}) \vee \ind(\Gamma_{n,5}).
\end{equation*}

For $n\geq 1$, $\ind(F_{n,5}) \simeq \Sigma\ind(B_{n-1,5}) \bigvee_2
\Sigma^2\ind(C_{n-1,5}) \bigvee
\Sigma^2\ind(D_{n-1,5}).$

in 1,2,...,5

[vertex] (1) at (2*,3*2) ;

in 1,2,...,5

[vertex] (2) at (2*,2*2) ;

in 2,...,5

[vertex] (3) at (2*,1*2) ;

in 2,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2,...,4

[edge] (1) – (1);

[evaluate=as using +1] in 1,2,...,4

[edge] (2) – (2);

[evaluate=as using +1] in 2,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 2,...,4

[edge] (4) – (4);

in 1,2,3,4,5

[edge] (1) – (2);

in 2,3,4,5

[edge] (2) – (3);

in 2,3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 2,4

[edge] (1) – (2);

[evaluate==int(1+)] in 2,4

[edge] (3) – (4);

[evaluate==int(-1)] in 3,5

[edge] (1) – (2);

[evaluate==int(-1)] in 3,5

[edge] (3) – (4);

[evaluate==int(1+)] in 1,3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
[label=[label distance=-3pt]-90:$g_{21}$] at (21) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;
[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]180:$g_{32}$] at (32) ;

$F_{n,5} \setminus N[f_2]$

in 1,2,...,5

[vertex] (1) at (2*,3*2) ;

in 3,...,5

[vertex] (2) at (2*,2*2) ;

in 3,...,5

[vertex] (3) at (2*,1*2) ;

in 2,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2,...,4

[edge] (1) – (1);

[evaluate=as using +1] in 3,...,4

[edge] (2) – (2);

[evaluate=as using +1] in 3,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 2,...,4

[edge] (4) – (4);

in 3,4,5

[edge] (1) – (2);

in 3,4,5

[edge] (2) – (3);

in 3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 2,4

[edge] (1) – (2);

[evaluate==int(1+)] in 4

[edge] (3) – (4);

[evaluate==int(-1)] in 5

[edge] (1) – (2);

[evaluate==int(-1)] in 3,5

[edge] (3) – (4);

[evaluate==int(1+)] in 3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 2,...,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[vertex] (f2) at (-2,4)  ;
[edge] (f2) – (11) ;

[label=[label distance=-3pt]180:$f_{1}$] at (f2) ;

[label=[label distance=-5pt]180:$g_{23}$] at (23) ;
[label=[label distance=-5pt]180:$g_{33}$] at (33) ;

$F_{n,5} \setminus N[g_{31}]$

in 1,2,...,5

[vertex] (1) at (2*,3*2) ;

in 1,2,...,5

[vertex] (2) at (2*,2*2) ;

in 2,...,5

[vertex] (3) at (2*,1*2) ;

in 3,...,5

[vertex] (4) at (2*,0*2) ;

[evaluate=as using +1] in 1,2,...,4

[edge] (1) – (1);

[evaluate=as using +1] in 1,2,...,4

[edge] (2) – (2);

[evaluate=as using +1] in 2,...,4

[edge] (3) – (3);

[evaluate=as using +1] in 3,...,4

[edge] (4) – (4);

in 1,2,3,4,5

[edge] (1) – (2);

in 2,3,4,5

[edge] (2) – (3);

in 3,4,5

[edge] (3) – (4);

[evaluate==int(1+)] in 2,4

[edge] (1) – (2);

[evaluate==int(1+)] in 2,4

[edge] (3) – (4);

[evaluate==int(-1)] in 3,5

[edge] (1) – (2);

[evaluate==int(-1)] in 5

[edge] (3) – (4);

[evaluate==int(1+)] in 1,3

[edge] (2) – (3);

[evaluate==int(-1)] in 4

[edge] (2) – (3);

[dotted, thick,shorten >=4pt, shorten <=4pt] (10,5) – (12,5);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,3) – (12,3);
[dotted, thick,shorten >=4pt, shorten <=4pt] (10,1) – (12,1);

[label=[label distance=-3pt]90:$g_{11}$] at (11) ;
[label=[label distance=-3pt]-90:$g_{21}$] at (21) ;
in 2,...,5

[label=[label distance=-3pt]90:$g_{1\x}$] at (1) ;

in 3,...,5

[label=[label distance=-3pt]-90:$g_{4\x}$] at (4) ;

[label=[label distance=-5pt]180:$g_{32}$] at (32) ;

[vertex] (f2) at (-2,4)  ;

[edge] (f2) – (21) ;
[edge] (f2) – (11) ;

[label=[label distance=-3pt]180:$f_{1}$] at (f2) ;

$F_{n,5} \setminus N[g_{41}]$

Since $f_3$ is a simplicial vertex and $N(f_3)=\{f_2,g_{31},g_{41}\}$ in the graph $F_{n,5}$ (see <Ref>), from <Ref>, we get

\begin{equation}{\label{Ln-sim-l3}}
\begin{split}
\ind(F_{n,5}) & \simeq \Sigma\ind(F_{n,5} \setminus \n[f_2]) \vee \Sigma\ind(F_{n,5} \setminus \n[g_{31}]) \vee \Sigma\ind(F_{n,5} \setminus \n[g_{41}]).
\end{split}
\end{equation}

Observe that the graph $F_{n,5}\setminus N[f_2]$ is isomorphic to $ B_{n-1,5}$ (see <Ref> and <Ref>). Hence, $\ind(F_{n,5}\setminus N[f_2])\simeq \ind(B_{n-1,5})$.

Since  $N(f_1) = \{g_{11}\}$  in $F_{n,5} \setminus N[g_{31}]$ (see <Ref>),   using <Ref>, we have $\ind(F_{n,5} \setminus N[g_{31}]) \simeq \Sigma \ind(F_{n,5} \setminus N[g_{31},g_{11}])$. However, the graph $F_{n,5} \setminus N[g_{31},g_{11}]$ is isomorphic to $ C_{n-1,5}$ and  therefore, $\ind(F_{n,5} \setminus N[g_{31}]) \simeq \Sigma \ind(C_{n-1,5})$.

Finally, consider the graph $F_{n,5} \setminus \n[g_{41}]$. Since $f_1$ is a simplicial vertex in $F_{n,5} \setminus \n[g_{41}]$, with $N(f_1)=\{g_{11},g_{21} \}$, from <Ref>, we have

\begin{equation}{\label{Ln-sim-l1}}
\begin{split}
\ind(F_{n,5} \setminus \n[g_{41}]) \simeq \Sigma\ind(F_{n,5} \setminus \n[g_{41},g_{11}]) \vee \Sigma \ind(F_{n,5} \setminus \n[g_{41},g_{21}]).
\end{split}
\end{equation}

Note that, the graphs $F_{n,5} \setminus \n[g_{41},g_{11}] $ and $F_{n,5} \setminus \n[g_{41},g_{21}] $ are isomorphic to  $D_{n-1,5}$ and $ C_{n-1,5}$, respectively (refer to <Ref>). Therefore, $\ind(F_{n,5} \setminus \n[g_{41}])\simeq \Sigma \ind(D_{n-1,5})\vee\Sigma \ind(C_{n-1,5})$. Now, the result follows from <Ref> by replacing all the associated homotopy equivalences.

For any graph $G_{n,5}\in \set{\Gamma_{n,5},\Lambda_{n,5}, A_{n,5}, B_{n,5}, C_{n,5}, D_{n,5}, E_{n,5}, F_{n,5}:  n\geq 0 }$, $\ind(G_{n,5})$ is homotopy equivalent to a wedge of spheres.

Since the graph  $\Gamma_{0,5}$ is a path on four vertices, $\ind( \Gamma_{0,5})$ is contractible.

We now compute $\ind(\Lambda_0)$.  Since $g_{11}$ is a simplicial vertex in the graph $\Lambda_{0,5}$, with $N(g_{11})=\{g_{21},l_{1},l_{2}\}$, using  <Ref>, we get

\begin{equation*}
\ind (\Lambda_{0,5})\simeq\Sigma(\ind(\Lambda_{0,5} \setminus \n[g_{21}])) \vee \Sigma(\ind(\Lambda_{0,5} \setminus \n[l_1])) \vee \Sigma(\ind(\Lambda_{0,5} \setminus \n[l_2])).
\end{equation*}

However, both the graphs $\Lambda_{0,5} \setminus \n[g_{21}]$ and $\Lambda_{0,5} \setminus \n[l_2]$ are isomorphic to the complete graph $K_3$ and the graph $\Lambda_{0,5} \setminus \n[l_1]$ is isomorphic to the complete graph $K_4$. Therefore,
$\ind (\Lambda_{0,5})\simeq\Sigma(\ind(K_3)) \vee \Sigma(\ind(K_3) )\vee \Sigma(\ind(K_4))
\simeq \vee_{7}\mathbb{S}^1.$

For any graph $G_{n, 5} \in \set{ A_{n,5}, B_{n,5}, C_{n,5}, D_{n,5}, E_{n,5}, F_{n,5} : n\geq 0}$, $\ind(G_{0, 5})$ is homotopy equivalent to a wedge of spheres from  homA basehomL base. Result follows from induction on $n$ and homG_n,5homFn.

§.§ Independence complex of Ind(Lambda n, for n >=0

Recall that for $n\geq 0$, the line graph of $P_{2n+3}\times P_5$ comprises of connected components $\Lambda_{n,5}$ and $\Tilde{\Gamma}_{n,5}$ (see <Ref>). The computation of $\ind(\Tilde{\Gamma}_{n,5})$ follows a similar approach as that of $\ind(\Gamma_{n,5})$. The intermediary graphs occurring during the computation of $\ind(\Tilde{\Gamma}_{n,5})$ resemble those defined in <Ref>, with the inclusion of certain additional vertices and edges.
For each $H_{n,5} \in \{\Gamma_{n,5},\Lambda_{n,5},A_{n,5},B_{n,5},C_{n,5},D_{n,5},E_{n,5},F_{n,5}\}$, where $n\in\mathbb{N}\cup\{0\}$, we define

\begin{equation*}
\begin{split}
V(\Tilde{H}_{n,5}) =  & V(H_{n,5})\cup \{g_{1\,2n+2},g_{2\,2n+2},g_{3\,2n+2},g_{4\,2n+2}\}, \\
E(\Tilde{H}_{n,5}) = & E(H_{n,5}) \cup \set{g_{i\,2n+1}g_{i\,2n+2} : i=1,2,3,4} \\
& \cup \set{g_{i\,2n+2}g_{i+1\,2n+2} : i = 1,2,3} \\
& \cup \set{g_{2\,2n+1}g_{3\,2n+2}, g_{3\,2n+1}g_{2\,2n+2}}.
\end{split}
\end{equation*}