# The distance spectral radius of $k$-uniform hypertrees with given number of
vertices of maximum degree. 111This work is partially supported by the
National Natural Science Foundation of China (No. 12271182).
E-mail<EMAIL_ADDRESS>(X<EMAIL_ADDRESS>(H. Shan)

Xiaoqi, Liu, Haiying Shan
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and
Applications
(Ministry of Education), Tongji University, Shanghai, China Corresponding
author

###### Abstract

This paper investigates the influence of two graft transformations on the
distance spectral radius of connected uniform hypergraphs. Specifically, we
study $k$-uniform hypertrees with given size, maximum degree and number of
vertices of maximum degree, and give the structure of such hypergraph with
maximum distance spectral radius.

Keywords: Distance spectral radius; Hypergraph; Graft transformation; Uniform
hypertree
2020 Mathematics Subject Classification: 05C50; 15A18

##  1 Introduction

A hypergraph $G$ consists of a nonempty vertex set $V(G)$ and an edge set
$E(G)$, where each edge $e\in E(G)$ is a nonempty subset of $V(G)$. The order
of $G$ is denoted as $|V(G)|$, and the size of $G$ is denoted as $|E(G)|$. For
an integer $k\geq 2$, a hypergraph $G$ is said to be $k$-uniform if every edge
has size $k$.

For $u,\ v\in V(G)$, a walk from $u$ to $v$ in $G$ is defined as a sequence of
vertices and edges
$(v_{0},e_{1},v_{1},\cdots,v_{p-1},e_{p},v_{p})$, where $v_{0}=u$ and
$v_{p}=v$. In this walk, each edge $e_{i}$ contains the vertices $v_{i-1}$ and
$v_{i}$, with $v_{i-1}\neq v_{i}$ for $i=1,\cdots,p$. The value $p$ represents
the length of this walk. A path is a walk where all $v_{i}$ and $e_{i}$ are
distinct. A cycle is a walk containing at least two distinct edges, with all
$e_{i}$ and $v_{i}$ distinct except for $v_{0}=v_{p}$. If there is a path from
$u$ to $v$ for any $u,\ v\in V(G)$, then we say that $G$ is connected. A
hypertree is a connected hypergraph with no cycles.

Let $G$ be a connected hypergraph. For $u,v\in V(G)$, the distance between $u$
and $v$ is defined as the length of the shortest path from $u$ to $v$ in $G$,
denoted as $d_{G}(u,v)$. Specifically, $d_{G}(u,u)=0$. The diameter of $G$ is
the maximum distance between all vertex pair of $G$. The distance matrix of
$G$ is the $n\times n$ matrix $D(G)=(d_{G}(u,v))_{u,v\in V(G)}$. The
eigenvalues of $D(G)$ are called the distance eigenvalues of $G$. Since $D(G)$
is real and symmetric, the distance eigenvalues of $G$ are real. The distance
spectral radius of $G$, denoted by $\rho(G)$, is the largest distance
eigenvalue of $G$. Note that $D(G)$ is an irreducible nonnegative matrix. The
Perron–Frobenius theorem implies that $\rho(G)$ is simple, and there is a
unique positive unit eigenvector that corresponds to $\rho(G)$, which is
called the distance Perron vector of $G$, denoted by $\mathbf{x}(G)$.

The study of distance eigenvalues has a rich history in graph theory. Its
origins can be traced back to the pioneering work of Graham and Pollack(1971)
[16] in which they described a relationship between the number of negative
distance eigenvalues and the addressing problem in data communication system.
Edelberg et al.[14]and Graham and Lovász[15] conducted early research on the
distance eigenvalues, laying the foundation for subsequent studies. Lin and
Zhou[19] determined the unique graphs with maximum distance spectral radius
among trees with given number of vertices of maximum degree. Lin and Zhou[17,
18] determined the the unique $k$-uniform hypertrees with the maximum, second
maximum, minimum, second minimum distance spectral radius. Moreover, they
presented the exceptional structure of k-uniform unicyclic hypergraphs that
have fixed sizes and possess minimum and second minimum distance spectral
radii, and provided the possible structure of the $k$-uniform unicyclic
hypergraph(s) of fixed size with maximum distance spectral radius. Liu et
al.[21] obtained the minimum distance spectral radius of $k$-uniform
hypertrees with given matching number. Wang and Zhou[23] determine the unique
graph that maximizes the distance spectral radius over all cacti with fixed
numbers of vertices and cycles in the context of hypertrees. For more
comprehensive discussions on distance eigenvalues, interested readers may
refer to the publications cited as references [13, 20, 24].

Let $\mathbb{T}_{k}(m,\Delta,n)$ be the set of $k$-uniform hypertrees of size
$m$ with $n$ vertices of maximum degree $\Delta$.

In this paper, motivated by the above results, we determine the unique
$k$-uniform hypertree with maximum distance spectral radius among $k$-uniform
hypertrees in $\mathbb{T}_{k}(m,\Delta,n)$ with $\Delta\geq 3$, which can be
seen as a generalization of the work in [19].

##  2 Notations and preliminaries

Let $G=(V,E)$ denote a (hyper)graph with vertex set $V$ and edge set $E$. For
any vertex of $G$, $N_{G}(v)$ denotes the set of vertices adjacent to $v$, and
$E_{G}(v)$ denotes the set of edges incident to $v$. The degree of a vertex
$v$ in $G$, denoted as $d_{G}(v)$, is the number of edges in $E_{G}(v)$. An
edge $e$ in hypergraph $G$ is called a pendant edge at vertex $u$ if $u$ is
the only vertex with degree greater than 1. The other vertices in $e$ are
called pendant vertices of $G$ or pendant neighbors of $u$.

A $k$-uniform hypertree with $m$ edges is a hyperstar, denoted by $S^{k}_{m}$,
if all edges share a common vertex. A $k$-uniform loose path with $m\geq 1$
edges, denoted by $P^{k}_{m}$, is the $k$-uniform hypertree whose vertices and
edges may be labeled as $(u_{0},e_{1},u_{1},\dots,u_{m-1},e_{m},u_{m})$ such
that the vertices $u_{1},\dots,u_{m-1}$ are of degree $2$, and all other
vertices of $G$ are of degree $1$.

For a hypergraph $G$ with vertex set $V(G)$ and edge set
$E(G)=\\{e_{1},\cdots,e_{m}\\}$, _weak vertices deletion_ of $U\subset V(G)$,
denoted by $G\backslash U$, creates the hypergraph
$G^{\prime}=(V(G^{\prime}),E(G^{\prime}))$ where $V(G^{\prime})=V(G)\setminus
U$ and $E(G^{\prime})=\\{e_{1}^{\prime},\cdots,e_{m}^{\prime}\\}$, such that
$e_{i}^{\prime}=e_{i}\setminus U$ for $1\leq i\leq m$. $G\backslash U$ is
written as $G\backslash\\{u\\}$ if $U=\\{u\\}$. For $e_{i}\in E(G)$,
$G^{\prime\prime}=G-e_{i}$ is the hypergraph with
$V(G^{\prime\prime})=V(G^{\prime\prime})$, $E(G^{\prime\prime})=E(G)\setminus
e_{i}$.

For a nonempty set $X\subseteq V(G)$, $G[X]$ represents the subhypergraph
induced by $X$, where $G[X]$ has vertex set $X$ and edge set of $\\{e\subseteq
X:e\in E(G)\\}$. Furthermore, $\sigma(X,\mathbf{x})$ denotes the sum of the
entries of the vector $\mathbf{x}$ corresponding to the vertices in $X$, if
$\mathbf{x}=\mathbf{x}(G)$, the alternative notation $\sigma_{G}(X)$ or
$\sigma(X)$ can be used for $\sigma(X,\mathbf{x})$,

Given a graph $G$ and a vertex $r\in V(G)$, the _rooted graph $G_{r}$_ is the
graph $G$ with the vertex $r$ labeled. Let $V^{\prime}(G)$ be the set of all
nonroot vertices of $G$, that is, $V^{\prime}(G)=V(G)\backslash\\{r\\}$.

Let $U=(v_{1},\cdots,v_{s})\in V(H)^{s}$ and $\mathcal{G}$ be a sequence of
$s$ rooted graphs $G_{r_{1}},G_{r_{2}},\cdots,G_{r_{s}}$ with root vertices
$r_{1},\dots,r_{s}$. The _rooted product_ of $H$ and $\mathcal{G}$, denoted
$H(U,\mathcal{G})$, is the graph formed by attaching each root of $G_{r_{i}}$
in $\mathcal{G}$ to its corresponding vertex $v_{i}$ in $H$. We denote
$G_{v_{i}}$ be the subhypergraph of $H(U,\mathcal{G})$ induced by
$(V(G_{r_{i}})\cup\\{v_{i}\\})\setminus\\{r_{i}\\}$ with root vertex $v_{i}$

Let $H=P_{m}(U,\mathcal{G})$ be a rooted product of the loose path
$P_{m}=(u_{0},e_{1},u_{1},\dots,u_{m-1},e_{m},u_{m})$ and the sequence of
rooted graphs $\mathcal{G}$. For any subset $Q$ of $U$, we write $H[Q]$ as a
subgraph of $H$ which is the rooted product of the subpath $P_{m}[Q]$ of
$P_{m}$ and the subsequence of $\mathcal{G}$ corresponding to $Q$, where
$P_{m}[Q]$ is the minimal subpath of $P_{m}$ that contains all vertices in
$Q$. Especially, we denote $H^{u_{i}}$ as the component of $H-e_{i+1}$
containing $u_{0}$, $H_{u_{i}}$ as the component of $H-e_{i}$ containing
$u_{d}$, respectively. We take $V^{\prime}(H_{u_{i}})=V(H_{u_{i}})\setminus
V(G_{u_{i}})$, $V^{\prime}(H^{u_{i}})=V(H^{u_{i}})\setminus V(G_{u_{i}})$. For
$0\leq c<d\leq m$, we denote $H_{u_{c}}^{u_{d}}$ as the subhypergraph of $H$
induced by $V(H)\setminus(V^{\prime}(H^{u_{c}})\cup V^{\prime}(H_{u_{d}})\cup
V^{\prime}(G_{u_{c}})\cup V^{\prime}(G_{u_{d}}))$.

With respect to the hypergraph mentioned above $H=P_{m}(U,\mathcal{G})$, if
each rooted graph $G_{r_{i}}$ in $\mathcal{G}$ is a hyperstar, we call it a
caterpillar. Furthermore, considering the situation where $U=\\{u_{i}\mid
0\leq i\leq m\\}$ and $\mathcal{G}=(G_{r_{i}}\mid 0\leq i\leq m)$, where the
rooted graph $G_{r_{i}}$ is a trivial graph for $i=0,m$ or $a<i<m-b$, and it
is $S_{\Delta-2}$ with its center as the root vertex for others, we can
rewrite $H$ as $C(m,\Delta,a,b)$. Furthermore, $C(m,\Delta,a,b)$ is denoted as
$C_{k}(m,\Delta,a,b)$ when $H$ is uniform with $k$.

Let $P_{s+t}^{k}=(u_{0},e_{1},u_{1},\cdots,u_{s+t-1},e_{s+t},u_{s+t})$ and
$H=P_{s+t}^{k}(U,\mathcal{G})$, where $U=(u_{1},\cdots,u_{s+t-1})$,
$\mathcal{G}=(G_{r_{1}},\cdots,G_{r_{s+t-1}})$, and rooted graphs in
$\mathcal{G}$ are $S_{c}^{k}$ with its center as the root vertex, except for
$G_{r_{s}}$. Denote the graph $H$ by $G_{c}(s,t)$.

Stevanović and Dragan investigated the adjacent spectral radius radius of
graphs that have path-like or star-like structures attached to them [22]. In
what follows, we will introduce the definition of hypergraph counterparts of
these graphs (refer to Fig. 1). Suppose $H$ is a connected hypergraph with
vertices $u,v\in V(H)$, and $G_{r}$ represents a rooted graph with vertex $r$
as its root. The hypergraph, $H_{u,v}(P_{s},P_{t})$, is constructed from $H$
by attaching two pendant hyperpaths, $P_{s}$ and $P_{t}$, of lengths $s$ and
$t$ to vertices $u$ and $v$, respectively. Furthermore, the hypergraph
$H_{u,v}(P_{s},P_{t},G_{r})$ is formed by identifying the root of a separate
copy of $G_{r}$ at each degree 2 vertex on the pendant hyperpaths $P_{s}$ and
$P_{t}$. If $u=v$, we condense the notation to $H_{u}(P_{s},P_{t},G_{r})$.
When $H,P_{s},P_{t},G_{r}$ are $k$ uniform hypergraphs,
$H_{u,v}(P_{s},P_{t},G_{r})$ is denoted by $H_{u,v}(s,t,G_{r})$. In
particular, if $G_{r}$ is a hyperstar with its center as the root vertex and
with size $c$, we denote $H_{u}(s,t,G_{r})$ and $H_{u,v}(s,t,G_{r})$ as
$H_{u}(s,t,c)$ and $H_{u,v}(s,t,c)$, respectively.

Let $H^{\prime}=H_{u,v}(P_{s},P_{t},G_{r})$, with $w\in V(H^{\prime})$ being
the pendant neighbor of a pendant vertex on subgraph $P_{t}$, and
$w^{\prime}\in V(H^{\prime})$ as a pendant vertex on subgraph $P_{s}$. The
hypergraph $H^{\prime\prime}$ emerges from $H^{\prime}$ by moving the edges in
$E_{v}(H^{\prime})$, excluding the non-pendant edge of $P_{t}$, from $w$ to
$w^{\prime}$. We say that $H^{\prime\prime}$ is obtained by graft
transformation.

$G_{r}$$G_{r}$$G_{r}$$G_{r}$$G_{r}$$G_{r}$$G_{r}$$\cdots$$P_{s}$$\cdots$$P_{t}$$v$$u$$H$
Figure 1: $H_{u,v}(P_{s},P_{t},G_{r})$

Lin and Zhou study the effect of some types of graft transformations to
increase or decrease the distance spectral radius of connected uniform
hypergraphs.[17]

######  Lemma 2.1.

[17] Let $H$ be a connected $k$-uniform hypergraph with $|E(H)|\geq 1$ and
$u\in V(H).$ For integers $s\geq t\geq 1$,
$\rho(H_{u}(s,t))<\rho(H_{u}(s+1,t-1)).$

######  Lemma 2.2.

[17] Let $H$ be a connected $k$-uniform hypergraph with $|E(H)|\geq 2$ and
$u,v\in e\in E(H)$. Suppose that $H-e$ consists of $k$ components and
$d_{H}(u)=d_{H}(v)=1$. For integers $s\geq t\geq 1$,
$\rho(H_{u,v}(s,t))<\rho(H_{u,v}(s+1,t-1)).$

For integer $m,a,b$, let $m^{\prime}=m-a-b-2$, $H=P_{m^{\prime}}^{k}$, $u,v$
be two pendant vertices of $H$ located in different pendant edges, then
$C_{k}(m,a,b,\Delta)$ can be rewritten as $H_{u,v}(a+1,b+1,\Delta-2)$. And the
alternative notation $G_{r_{s}}(s,t,c)$ can be used for $G_{c}(s,t)$.

In Sections 3 and 4, we investigated the effects of graft transformations in
two specific cases on the distance spectral radius of connected uniform
hypergraphs. Based on these results, we determine the structure of the
hypertree in $\mathbb{T}_{k}(m,\Delta,n)$ with maximum distance spectral
radius.

Consider a connected $k$-uniform hypergraph $G$ and $\mathbf{x}$ be a real
column vector whose coordinates are indexed by $V(G)$. For $U\subseteq V(G)$,
$v\in V(G)$, we define
$W(U,v,\mathbf{x})=\displaystyle\sum_{\begin{subarray}{c}u\in
U_{1}\end{subarray}}x_{u}d(u,v)$, where $x_{u}$ refers to the $\mathbf{x}$
coordinate corresponding to vertex $u$. If $H$ is a subgraph of $G$, the
alternative notation $W(H,v,\mathbf{x})$ can be used for
$W(V(H),v,\mathbf{x})$. If $\mathbf{x}$ is the distance Perron vector of $H$,
the expressions $W(U,v,\mathbf{x})$, $W(V(H),v,\mathbf{x})$, and
$W(H,v,\mathbf{x})$ can be simplified to $W(U,v)$, $W(V(H),v)$, and $W(H,v)$,
respectively. If $\mathbf{x}$ is an all-ones vector, the expressions
$W(U,v,\mathbf{x})$, $W(V(H),v,\mathbf{x})$, and $W(H,v,\mathbf{x})$ can be
rewritten as $W_{0}(U,v)$, $W_{0}(V(H),v)$, and $W_{0}(H,v)$, respectively.
And for $U_{1},U_{2}\subseteq V(G)$, we define
$W(U_{1},U_{2})=\displaystyle\sum_{\begin{subarray}{c}u\in U_{1};v\in
U_{2}\end{subarray}}x_{u}d(u,v)$, where $x_{u}$ refers to the $\mathbf{x}(G)$
coordinate corresponding to vertex $u$. For $U,V_{1},V_{2}\subseteq V(G)$,
denote $W(U,V_{1})-W(U,V_{2})$ as $WD(U;V_{1},V_{2})$.

If $G=G_{v_{1}}(w)G_{v_{2}}$, then $w$ is a cut vertex of $G$. We have that:
$W(G_{v_{1}},u)=W(G_{v_{1}},w)+d(w,u)\sigma(G_{v_{1}})$ for any vertex of
$G_{v_{2}}$.

Then

$\mathbf{x}^{T}D(G)\mathbf{x}=\sum_{\\{u,v\\}\subseteq
V(G)}2d_{G}(u,v)x_{u}x_{v}=\sum_{u\in V(G)}x_{u}W(G,u,\mathbf{x}),$

and $\rho$ is a distance eigenvalue with corresponding eigenvector
$\mathbf{x}$ if and only if $\mathbf{x}\neq 0$ and for each $u\in V(G)$,

$\displaystyle\rho x_{u}=\sum_{v\in V(G)}d_{G}(u,v)x_{v}=W(G,u,\mathbf{x}).$
(1)

This equation is known as the eigenequation for $\rho(G)$ at $u$. By
Rayleigh’s principle, for a unit column vector $\mathbf{x}\in\mathbb{R}^{n}$,
we have $\rho(G)\geq\mathbf{x}^{T}D(G)\mathbf{x}$, with equality if and only
if $\mathbf{x}=\mathbf{x}(G)$.

We say that two vertices $u,v\in V$ in a hypergraph $G=(V,E)$ are similar if
there is a graph automorphism $\phi\in Aut(G)$, s.t. $\phi(u)=v$.

######  Lemma 2.3.

[17] Let $u,v$ be similar vertices in hypergraph $G$ and
$\mathbf{x}=\mathbf{x}(G)$. Then $x_{u}=x_{v}$.

Let $u^{\prime}$ be a pendant neighbor of $u$ in $k$-unifrom hypergraph $G$
with $u,u^{\prime}\in e$. Define $G^{\prime}=G-U$ where $U=e-\\{u\\}$.
Consider the eigenequations for $\rho(G)$ at $u$ and $u^{\prime}$.

$\displaystyle\rho(G)x_{u}=$ $\displaystyle
W(G^{\prime},u)+(k-1)x_{u^{\prime}}$ (2) $\displaystyle\rho(G)x_{u^{\prime}}=$
$\displaystyle
W(G^{\prime},u^{\prime})+(k-2)x_{u^{\prime}}=W(G^{\prime},u)+\sigma(G^{\prime})+(k-2)x_{u^{\prime}}$
$\displaystyle=$ $\displaystyle W(G^{\prime},u)+\sigma(G)-x_{u^{\prime}}.$ (3)

From (2) and (3) and Lemma 2.3 we have:

$(\rho(G)+k)x_{u^{\prime}}-\rho(G)x_{u}=\sigma(G)$ (4)

Let $u$ be a vertex of degree one in $e\in G$. For any $v\in e$, we have
$d_{G}(u,w)\geq d_{G}(v,w)$ for $w\in V(G)\backslash\\{u\\}$. So
$(\rho(G)+1)(x_{u}-x_{v})\geq 0$ and $x_{u}\geq x_{v}$, where the equality
holds if only if $d_{G}(v)=1$.

Let $H$ be a connected uniform hypergraph obtained from the hypergraph $G$ by
identifying two $u_{1}$ and $u_{2}$ of $G$ with the center vertex of
$k$-uniform hyperstars $S_{1}$ and $S_{2}$ with size $l\geq 0$, respectively.
Let $u^{\prime}_{1},u^{\prime}_{2}$ be the pendant neighbor of $u_{1},u_{2}$,
respectively. Then
$\sigma(S_{1})-\sigma(S_{2})=x_{u_{1}}-x_{u_{2}}+l(k-1)(x_{u^{\prime}_{1}}-x_{u^{\prime}_{2}})$.
From Eq. (4), we obtain:

$\displaystyle\sigma(S_{1})-\sigma(S_{2})=$
$\displaystyle\bigg{(}\frac{\rho(T)l(k-1)}{\rho(T)+k}+1\bigg{)}(x_{u_{1}}-x_{u_{2}}).$

Therefore,

$\operatorname{sgn}(\sigma(S_{1})-\sigma(S_{2}))=\operatorname{sgn}(x_{u_{1}}-x_{u_{2}}).$
(5)

Let $G$ be a $k$-uniform hypergraph with $u,v\in V(G)$ and
$e_{1},\cdots,e_{r}\in E(G)$ such that $u\notin e_{i}$ and $v\in e_{i}$ for
$1\leq i\leq r$. Let $e_{i}^{\prime}=(e_{i}\setminus\\{v\\})\cup\\{u\\}$ for
$1\leq i\leq r$. Let $G^{\prime}$ be the hypergraph with $V(G^{\prime})=V(G)$
and
$E(G^{\prime})=(E(G)\setminus\\{e_{1},\cdots,e_{r}\\})\cup\\{e^{\prime}_{1},\cdots,e^{\prime}_{r}\\}.$
Then we say that $G^{\prime}$ is obtained from $G$ by moving edges
$e_{1},...,e_{r}$ from $v$ to $u$.

Let $u_{1},u_{2}\in V(H_{0})$ and $r$ be the rooted vertex of $G$. Take
$H_{1}=H_{0}(u_{1},G)$, $H_{2}=H_{0}(u_{2},G)$. Then

$\displaystyle\frac{1}{2}\mathbf{x}^{T}(D(H_{2})-D(H_{1}))\mathbf{x}=$
$\displaystyle\sigma(V^{\prime}(G),\mathbf{x})(W(H_{0},u_{2};\mathbf{x})-W(H_{0},u_{1};\mathbf{x}))$
(6)

Take $\mathbf{x}=\mathbf{x}(H_{1})$. Then

$\displaystyle\rho(H_{1})(x_{u_{2}}-x_{u_{1}})=$ $\displaystyle
d(u_{1},u_{2})\sigma(V^{\prime}(G),\mathbf{x})+(W(H_{0},u_{2};\mathbf{x})-W(H_{0},u_{1};\mathbf{x}))$
(7)

So

$\rho(H_{2})-\rho(H_{1})\geq\sigma(V^{\prime}(G),\mathbf{x})(\rho(H_{1})(x_{u_{2}}-x_{u_{1}})-d(u_{1},u_{2})\sigma(V^{\prime}(G),\mathbf{x})).$

Let $u_{1},u_{2}\in V(H_{0})$ and $r_{1},r_{2}$ be the rooted vertex of
$G_{r_{1}},\ G_{r_{2}}$, respectively. Take
$H_{1}=H_{0}(u_{1},u_{2},G_{r_{1}},G_{r_{2}})$,
$H_{2}=H_{0}(u_{2},u_{1},G_{r_{2}},G_{r_{2}})$. Then by using Eq. (6) twice,
we have:

$\displaystyle\frac{1}{2}\mathbf{x}^{T}(D(H_{2})-D(H_{1}))\mathbf{x}=$
$\displaystyle(\sigma(V^{\prime}(G_{u_{1}}))-\sigma(V^{\prime}(G_{u_{2}})))(W(H_{0},u_{2};\mathbf{x})-W(H_{0},u_{1};\mathbf{x}))$
(8)

######  Lemma 2.4.

Suppose that $u_{1},u_{2}$ are two distinct vertices of $H$, and $r_{1},r_{2}$
are rooted vertices of nontrivial rooted graphs $G_{r_{1}}$ and $G_{r_{2}}$,
respectively. Take
$H_{1}=H((u_{1},u_{1}),\mathcal{G}),H_{2}=H((u_{1},u_{2}),\mathcal{G})$. Then
$\rho(H_{2})>\rho(H_{1})$ holds if one of the following two conditions is
satisfied:

  1. (1).

$d_{H}(u_{1})=d_{H}(u_{2})=d_{H}(u_{1},u_{2})=1$;

  2. (2).

$\sigma_{H_{1}}(V^{\prime}(G_{r_{1}}))+x_{u_{1}}\geq\sigma_{H_{1}}(H)$

###### Proof.

Take $G_{0}=H(u_{1},G_{r_{1}})$, then $H_{1}=G_{0}(u_{1},G_{r_{2}})$ and
$H_{2}=G_{0}(u_{2},G_{r_{2}})$. Let $\mathbf{x}=\mathbf{x}(H_{1})$. By Eq.
(6), we have

$\displaystyle\frac{1}{2}(\rho(H_{2})-\rho(H_{1}))\geq$
$\displaystyle\frac{1}{2}\mathbf{x}^{T}(D(H_{2})-D(H_{1}))\mathbf{x}$
$\displaystyle=$
$\displaystyle\sigma(V^{\prime}(G_{r_{2}}))(W(G_{0},u_{2})-W(G_{0},u_{1}))$
$\displaystyle=$
$\displaystyle\sigma(V^{\prime}(G_{r_{2}}))(d_{H}(u_{1},u_{2})\sigma(V^{\prime}(G_{r_{1}}))+W(H,u_{2})-W(H,u_{1}))$

(1). If $d_{H}(u_{1})=d_{H}(u_{2})=d_{H}(u_{1},u_{2})=1$, then
$W(H,u_{1})-W(H,u_{2})=0$ and $\frac{1}{2}(\rho(H_{2})-\rho(H_{1}))>0$ follow.
(2). When $\sigma_{H_{1}}(V^{\prime}(G_{r}))+x_{u}\geq\sigma_{H_{1}}(H)$, we
have:

$\displaystyle W(H,u_{2})-W(H,u_{1})$
$\displaystyle=d(u_{1},u_{2})x_{u_{1}}+W(H\setminus\\{u_{1}\\},u_{2})-W(H\setminus\\{u_{1}\\},u_{1})$
$\displaystyle\geq
d(u_{1},u_{2})x_{u_{1}}-d(u_{1},u_{2})\sigma_{H_{1}}(H\setminus\\{u_{1}\\})>d(u_{1},u_{2})x_{u_{1}}-d(u_{1},u_{2})\sigma_{H_{1}}(H),$

then $\frac{1}{2}(\rho(H_{2})-\rho(H_{1}))>0$.

∎

For $H=P_{m}^{k}(U,\mathcal{G})$ where
$P_{m}^{k}=(u_{0},e_{1},u_{1},\cdots,u_{m-1},e_{m},u_{m})$,
$U=(u_{1},\cdots,u_{m-1})$, $\mathcal{G}=(G_{r_{1}},\cdots,G_{r_{m-1}})$.
Taking $\mathbf{x}=\mathbf{x}(H)$, by Lemma 2.3, we can use $x_{w_{i}}$ to
donate the component of $\mathbf{x}$ that corresponds to the vertices in
$e_{i}$ of degree $1$.

For $1\leq i\leq m^{*}-1$, we have:

$\displaystyle\rho(H)x_{u_{i}}=$ $\displaystyle
W(H^{u_{i-1}},u_{i})+W(H_{u_{i}},u_{i})+(k-2)x_{w_{i}}$
$\displaystyle\rho(H)x_{w_{i}}=$ $\displaystyle
W(H^{u_{i-1}},w_{i})+W(H_{u_{i}},w_{i})+(k-3)x_{w_{i}}$ $\displaystyle=$
$\displaystyle
W(H^{u_{i-1}},u_{i})+W(H_{u_{i}},u_{i})+\sigma(H_{u_{i}})+(k-3)x_{w_{i}}$
$\displaystyle=$
$\displaystyle\rho(H)x_{u_{i}}+\sigma(H_{u_{i}})-x_{w_{i}}=\rho(H)x_{u_{i-1}}+\sigma(H^{u_{i-1}})-x_{w_{i}}$
(9)

From Eq. (9), we have:

$\displaystyle\rho(H)(x_{u_{i-1}}-x_{u_{i}})=$
$\displaystyle\sigma(H_{u_{i}})-\sigma(H^{u_{i-1}})$ (10)
$\displaystyle(\rho(H)+1)(x_{w_{i}}-x_{w_{i+1}})=$
$\displaystyle\sigma(H_{u_{i}})-\sigma(H^{u_{i}})$ (11)
$\displaystyle\rho(H)(x_{u_{i}}+x_{u_{i-1}})=$
$\displaystyle(2\rho(H)+k)x_{w_{i}}-\sigma(H)$ (12)

From Eq. (10) and Eq. (12), we have:

$\displaystyle\rho(H)(x_{u_{i-1}}-x_{u_{i+1}})=$
$\displaystyle(2\rho(H)+k)(x_{w_{i}}-x_{w_{i+1}})$ $\displaystyle=$
$\displaystyle\frac{2\rho(H)+k}{\rho(H)+k-1}(\sigma(H_{u_{i+1}})-\sigma(H^{u_{i-1}})).$
(13)

Take integers $s,t$ such that $s+t=m_{0},1\leq s-t\leq 2$. From (10), we have:

$\displaystyle\rho(H)((x_{u_{t-i}}-x_{u_{s+i}})-(x_{u_{t+1-i}}-x_{u_{s+i-1}}))=\sigma(H_{u_{t+1-i}})-\sigma(H^{u_{t-i}})+\sigma(H_{u_{s+i}})-\sigma(H^{u_{s+i-1}})$
$\displaystyle=$ $\displaystyle
2(\sigma(H_{u_{s+i}})-\sigma(H^{u_{t-i}}))+(k-2)(x_{w_{s+i}}-x_{w_{t+1-i}})$
(14) $\displaystyle=$ $\displaystyle
2(\sigma(H_{u_{s}})-\sigma(H^{u_{t}}))+2(\sigma(H_{u_{s+i}})-\sigma(H_{u_{s}})+\sigma(H^{u_{t}})-\sigma(H^{u_{t-i}}))+(k-2)(x_{w_{s+i}}-x_{w_{t+1-i}})$
$\displaystyle=$

$2(\sigma(H_{u_{s}})-\sigma(H^{u_{t}}))+2\sum_{l=0}^{i-1}(\sigma(G_{u_{t-l}})-\sigma(G_{u_{s+l}})+(k-2)(x_{w_{t-l}}-x_{w_{s+1+l}}))+(k-2)(x_{w_{s+i}}-x_{w_{t+1-i}}).$

(15)

According to (11), we have:

$\displaystyle(\rho(H)+1)\big{(}(x_{w_{t-i+1}}-x_{w_{s+i}})-(x_{w_{t+2-i}}-x_{w_{s+i-1}})\big{)}$
$\displaystyle=$ $\displaystyle
2(\sigma(H_{u_{s+i-1}})-\sigma(H^{u_{t-i+1}}))+\sigma(G_{t-i+1})-\sigma(G_{s+i-1})$
(16) $\displaystyle=$ $\displaystyle
2(\sigma(H_{u_{s}})-\sigma(H^{u_{t}}))+2(\sigma(H_{u_{s+i-1}})-\sigma(H_{u_{s}})+\sigma(H^{u_{t}})-\sigma(H^{u_{t-i+1}}))+\sigma(G_{u_{t-i+1}})-\sigma(G_{u_{s+i-1}})$
$\displaystyle=$ $\displaystyle
2(\sigma(H_{u_{s}})-\sigma(H^{u_{t}}))+2\sum_{l=0}^{i-2}(\sigma(G_{u_{t-l}})-\sigma(G_{u_{s+l}})+(k-2)(x_{w_{t-l}}-x_{w_{s+1+l}}))+\sigma(G_{u_{t-i+1}})-\sigma(G_{u_{s+i-1}}).$
(17)

######  Lemma 2.5.

Suppose $1\leq s-t\leq 2$ and
$\operatorname{sgn}(\sigma(G_{u_{t-l}})-\sigma(G_{u_{s+l}}))=\operatorname{sgn}(x_{u_{t-l}}-x_{u_{s+l}})$
for $0\leq l\leq r$. Then
(1). $x_{u_{t-l}}-x_{u_{s+l}}$ and $\sigma(H_{u_{s}})-\sigma(H^{u_{t}})$ have
common sign for $0\leq l\leq r+1$.
(2). $x_{w_{t+1}}-x_{w_{s}}$ and $\sigma(H_{u_{s}})-\sigma(H^{u_{t}})$ have
common sign or $x_{w_{t+1}}-x_{w_{s}}=0$. $x_{w_{t+1-l}}-x_{w_{s+l}}$ and
$\sigma(H_{u_{s}})-\sigma(H^{u_{t}})$ have common sign for $1\leq l\leq r+1$.

###### Proof.

From Eq. (10) and (13), we get (1) is true for $l=0$ and
$x_{w_{t+1}}-x_{w_{s}}$ and $\sigma(H_{u_{s}})-\sigma(H^{u_{t}})$ have common
sign or $x_{w_{t+1}}-x_{w_{s}}=0$.

Suppose that for $0\leq l\leq m-1\leq r$, $x_{u_{t-l}}-x_{u_{s+l}}$ and
$\sigma(H_{u_{s}})-\sigma(H^{u_{t}})$ have common sign,
$x_{w_{t+1}}-x_{w_{s}}$ and $\sigma(H_{u_{s}})-\sigma(H^{u_{t}})$ have common
sign or $x_{w_{t+1}}-x_{w_{s}}=0$, and for $1\leq l\leq m-1\leq r$,
$x_{w_{t+1-l}}-x_{w_{s+l}}$ and $\sigma(H_{u_{s}})-\sigma(H^{u_{t}})$ have
common sign , from Eq. (15) and (17), we have $x_{u_{t-m}}-x_{u_{s+m}}$ ,
$x_{w_{t+1-m}}-x_{w_{s+m}}$ and $\sigma(H_{u_{s}})-\sigma(H^{u_{t}})$ have
common sign. By induction hypothesis, we complete the proof. ∎

From Eq. (15) and (17) and Lemma 2.5, we have:

######  Corollary 2.1.

Suppose $1\leq s-t\leq 2$, $\sigma(H_{u_{s}})-\sigma(H^{u_{t}})>0$ and
$\operatorname{sgn}(\sigma(G_{u_{t-l}})-\sigma(G_{u_{s+l}}))=\operatorname{sgn}(x_{u_{t-l}}-x_{u_{s+l}})$
for $0\leq l\leq r$. Then we have :
$x_{u_{t-l}}-x_{u_{s+l}}>x_{u_{t-l+1}}-x_{u_{s+l-1}}>0$ and
$x_{w_{t-l+1}}-x_{w_{s+l}}>x_{w_{t-l+2}}-x_{w_{s+l-1}}\geq 0$ for $0\leq l\leq
r+1$.

######  Corollary 2.2.

Suppose $1\leq s-t\leq 2$, $\sigma(H^{u_{t}})-\sigma(H_{u_{s}})>0$ and
$\operatorname{sgn}(\sigma(G_{u_{t-l}})-\sigma(G_{u_{s+l}}))=\operatorname{sgn}(x_{u_{t-l}}-x_{u_{s+l}})$
for $0\leq l\leq r$. Then we have :
$x_{u_{t-l}}-x_{u_{s+l}}<x_{u_{t-l+1}}-x_{u_{s+l-1}}<0$ and
$x_{w_{t-l+1}}-x_{w_{s+l}}<x_{w_{t-l+2}}-x_{w_{s+l-1}}\leq 0$ for $0\leq l\leq
r+1$.

######  Lemma 2.6.

Let $H=G_{c}(s,t)$. For $s\geq t\geq 2$, we have
$\sigma(V^{\prime}(H^{u_{s}}))+x_{u_{s}}>\sigma(V^{\prime}(H_{u_{s+1}}))+x_{u_{s+1}}$,
$\sigma(V^{\prime}(H^{u_{s}}))\geq\sigma(V^{\prime}(H_{u_{s}}))$.

###### Proof.

By Lemma 2.5, we know that $\sigma(H^{u_{s-1}})-\sigma(H_{u_{s+1}})$ and
$x_{u_{s+i}}-x_{u_{s-i}},x_{w_{s+i}}-x_{w_{s+1-i}}$ have common sign for
$1\leq i\leq t$. Note that

$\sigma(H^{u_{s-1}})-\sigma(H_{u_{s+1}})\geq-(\sum_{i=1}^{t}(\sigma(G_{u_{s+i}})-\sigma(G_{u_{s-i}}))+(k-2)\sum_{i=2}^{t}(x_{w_{s+i}}-x_{w_{s+1-i}}))$

By (5), $\sigma(H_{s+i})-\sigma(H_{s-i})$ and $x_{u_{s+i}}-x_{u_{s-i}}$ have
common sign, which implies $\sigma(H^{u_{s-1}})\geq\sigma(H_{u_{s+1}})$, then
$\sigma(V^{\prime}(H^{u_{s}}))+x_{u_{s}}>\sigma(H^{u_{s-1}})\geq\sigma(H_{u_{s+1}})>\sigma(V^{\prime}(H_{u_{s+1}}))+x_{u_{s+1}}$.

By
$x_{u_{s+1}}-x_{u_{s}}=\frac{2\rho(G)+k}{\rho(G)}(x_{w_{s+1}}-x_{w_{s}})\geq
0$. Note that:

$\rho(H)(x_{u_{s+1}}-x_{u_{s-1}})=2(\sigma(V^{\prime}(H^{u_{s}}))-\sigma(V^{\prime}(H_{u_{s}})))+(k-2)(x_{w_{s+1}}-x_{w_{s}}),$

and $\rho(G)>k$, we have
$\sigma(V^{\prime}(H^{u_{s}}))\geq\sigma(V^{\prime}(H_{u_{s}}))$. ∎

##  3 Distance spectral radius of caterpillars in $\mathbb{T}_{k}(m,\Delta,n)$

In this section, we investigate the distance spectral radius of the
caterpillars in $T=C_{k}(m^{*},\Delta,a,b)$, and we give the structure of the
caterpillars in $\mathbb{T}_{k}(m,\Delta,n)$ with the maximum distance
spectral radius. Taking $\mathbf{x}=\mathbf{x}(T)$, by Lemma 2.3, we can use
$x_{v_{i}}$ to donate the component of $\mathbf{x}$ that corresponds to the
pendant neighbor of $u_{i}$, and $x_{w_{i}}$ to donate the component of
$\mathbf{x}$ that corresponds to the vertices of $e_{i}$ of degree $1$.Denote
$G_{u_{i}}$ as $S_{i}$.

These are some important facts about the components of the vector
$\mathbf{x}=\mathbf{x}(T)$.

######  Fact 1.

Let integers $a,b,m^{*}$ as defined above. Then we have

$0<x_{u_{0}}-x_{u_{m^{*}}}<x_{u_{1}}-x_{u_{m^{*}-1}}<\cdots<x_{u_{a}}-x_{u_{m^{*}-a}}<x_{u_{a+1}}-x_{u_{m^{*}-a-1}}$

and

$0<x_{w_{1}}-x_{w_{m^{*}}}<x_{w_{2}}-x_{w_{m^{*}-1}}<\cdots<x_{w_{a+1}}-x_{w_{m^{*}-a}}<x_{u_{a+1}}-x_{u_{m^{*}-a-1}}.$

###### Proof.

Take integers $p+q=m^{*}$, $1\leq q-p\leq 2$. From $0\leq a\leq b-2$, we know
that $S_{p-j}\cong S_{q+j}$ or $S_{q+j}\cong S^{k}_{\Delta-2}$ and $S_{p-j}$
is trivial. For $0\leq j\leq p$, if $x_{u_{p-j}}\leq x_{u_{q+j}}$, then
$\sigma(S_{p-j})\leq\sigma(S_{q+j})$ by (5).

Suppose $\sigma(T_{u_{q}})-\sigma(T^{u_{p}})\leq 0$, by Lemma 2.5, we have
$x_{u_{p-i}}\leq x_{u_{q+i}},x_{w_{p+1-i}}\leq x_{w_{q+i}}$ for all $0\leq
i\leq p$. Since $b\geq a+2$, there exists an $0\leq i_{0}\leq p$ such that
$\sigma(S_{i_{0}})<\sigma(S_{m^{*}-1-i_{0}})$. Then we have
$\sigma(T_{u_{q}})>\sigma(T^{u_{p}})$, a contradiction. Therefore,

$\sigma(T_{u_{q}})-\sigma(T^{u_{p}})>0.$

From Eq. (10) and (13) and $1\leq q-p\leq 2$, we have $x_{u_{p}}>x_{u_{q}}$
and $x_{w_{p+1}}\geq x_{w_{q}}$.

Suppose $x_{u_{0}}\leq x_{u_{m^{*}}}$. Then we can show $x_{w_{1}}\leq
x_{w_{m^{*}}}$ due to $x_{u_{0}}=x_{w_{1}}$ and $x_{u_{m^{*}}}=x_{w_{m^{*}}}$.

For $p+q=m^{*}$, by Eq. (14) and (16) we have:
If $x_{u_{j}}\leq x_{u_{m^{*}-j}}\text{ and }x_{w_{j}}\leq x_{w_{m*-j+1}}$ for
$0\leq j\leq i$, then
$x_{u_{i+1}}-x_{u_{m^{*}-i-1}}\leq x_{u_{i}}-x_{u_{m^{*}-i}}\leq 0$,
$x_{w_{i+1}}-x_{w_{m^{*}-i}}\leq x_{w_{i}}-x_{w_{m^{*}-i+1}}\leq 0$.
So $x_{u_{i}}\leq x_{u_{m^{*}-i}}$ for $0\leq i\leq p$. It follows that
$x_{u_{p}}\leq x_{u_{q}}$, a contradiction.
Thus $x_{u_{0}}>x_{u_{m^{*}}}$ and $x_{w_{1}}>x_{w_{m^{*}}}$.

From $S_{i}\cong S_{m^{*}-i}$ for $1\leq i\leq a$, we have
$\operatorname{sgn}(\sigma(S_{i})-\sigma(S_{m^{*}-i}))=x_{u_{i}}-x_{u_{m^{*}-i}}$
for $1\leq i\leq a$. By Eq. (14) and (16), take $s=l+1,t=m^{*}-l-1$, we have
for $0\leq l\leq a$,
$x_{u_{l+1}}-x_{u_{m^{*}-l-1}}>x_{u_{l}}-x_{u_{m^{*}-l}}>0$ and for $1\leq
k\leq a$, $x_{w_{k+1}}-x_{w_{m^{*}-k}}>x_{w_{k}}-x_{w_{m^{*}-k+1}}>0$ by
induction on $l$.

Note that $\sigma(T_{u_{a+1}})=\sigma(T)-\sigma(V^{\prime}(T^{u_{a+1}}))$,
$\sigma(T^{u_{m^{*}-a-1}})=\sigma(T)-\sigma(V^{\prime}(T_{m^{*}-a-1}))$. By
$\operatorname{sgn}(\sigma(S_{i})-\sigma(S_{m^{*}-i}))=x_{u_{i}}-x_{u_{m^{*}-i}}\
\text{for}\ 1\leq i\leq a,$

we have $\sigma(V^{\prime}(T^{u_{a+1}}))>\sigma(V^{\prime}(T_{m^{*}-a-1}))$,
then we have

$\sigma(T_{u_{a+1}})<\sigma(T^{u_{m^{*}-a-1}}).$

From Eq. (9), we have

$(\rho(T)+1)x_{w_{a+1}}-\rho(T)x_{u_{a+1}}=\sigma(T_{u_{a+1}})<\sigma(T^{u_{m^{*}-a-1}})=(\rho(T)+1)x_{w_{m^{*}-a}}-\rho(T)x_{u_{m^{*}-a-1}}.$

So
$\rho(T)(x_{u_{a+1}}-x_{u_{m^{*}-a-1}})>(\rho(T)+1)(x_{w_{a+1}}-x_{w_{m^{*}-a}})$.

Thus $x_{u_{a+1}}-x_{u_{m^{*}-a-1}}>x_{w_{a+1}}-x_{w_{m^{*}-a}}>0$. ∎

######  Fact 2.

Let $m^{\prime}=m^{*}+a-b+1$, $p^{\prime}+q^{\prime}=m^{\prime}$ and $1\leq
q^{\prime}-p^{\prime}\leq 2$. Then we have:

$0<x_{u_{p^{\prime}}}-x_{u_{q^{\prime}}}<x_{u_{p^{\prime}-1}}-x_{u_{q^{\prime}+1}}<\cdots<x_{u_{a+1}}-x_{u_{m^{*}-b}}$

and

$0<x_{w_{p^{\prime}+1}}-x_{w_{q^{\prime}}}<x_{w_{p^{\prime}}}-x_{w_{q^{\prime}+1}}<\cdots<x_{w_{a+2}}-x_{w_{m^{*}-b}}<x_{u_{a+1}}-x_{u_{m^{*}-b}}.$

###### Proof.

From $a+2\leq b,1\leq q-p\leq 2$ and $p+q=m^{*}$, we have $p^{\prime}\leq p$
and $q^{\prime}\leq q$.

By Fact 1, we have
$\sigma(T^{u_{p^{\prime}}})\leq\sigma(T^{u_{p}})<\sigma(T_{u_{q}})\leq\sigma(T_{u_{q^{\prime}}})$.

From Lemma 2.5 and Corollary 2.1, we have
$x_{u_{i}}-x_{u_{m^{\prime}-i}}>x_{u_{i+1}}-x_{u_{m^{\prime}-i-1}}>0,$
$x_{w_{i+1}}-x_{w_{m^{\prime}-i}}>x_{w_{i+2}}-x_{w_{m^{\prime}-i-1}}>0$ for
$a+1\leq i\leq p^{\prime}-1$.

Then we have
$\sigma(V^{\prime}(T_{u_{a+1}}^{u_{p^{\prime}}}))+x_{u_{p^{\prime}}}>\sigma(V^{\prime}(T_{q^{\prime}}^{u_{m^{*}-b}}))+x_{u_{q^{\prime}}}$.

Note that

$\sigma(T^{u_{a+1}})+\sigma(V^{\prime}(T_{u_{a+1}}^{u_{p^{\prime}}}))+x_{u_{p^{\prime}}}=\sigma(T^{p^{\prime}})<\sigma(T_{u_{q^{\prime}}})=\sigma(T_{u_{m^{*}-b}})+\sigma(V^{\prime}(T_{q^{\prime}}^{u_{m^{*}-b}}))+x_{u_{q^{\prime}}}.$

we have

$\sigma(T^{u_{a+1}})<\sigma(T_{u_{m^{*}-b}}).$

From Eq. (9), we have

$(\rho(T)+1)x_{w_{a+2}}-\rho(T)x_{u_{a+1}}=\sigma(T^{u_{a+1}})<\sigma(T_{u_{m^{*}-b}})=(\rho(T)+1)x_{w_{m^{*}-b}}-\rho(T)x_{u_{m^{*}-b}}.$

So
$\rho(T)(x_{u_{a+1}}-x_{u_{m^{*}-b}})>(\rho(T)+1)(x_{w_{a+2}}-x_{w_{m^{*}-b}})$.

Thus $x_{u_{a+1}}-x_{u_{m^{*}-b}}>x_{w_{a+2}}-x_{w_{m^{*}-b}}>0$. ∎

######  Fact 3.

Let $m^{\prime\prime}=2m^{*}-a-b-1$,
$p^{\prime\prime}+q^{\prime\prime}=m^{\prime\prime}$ and $1\leq
q^{\prime\prime}-p^{\prime\prime}\leq 2$. Then we have

$x_{u_{m^{*}-b}}-x_{u_{m^{*}-a-1}}<x_{u_{m^{*}-b+1}}-x_{u_{m^{*}-a-2}}<\cdots<x_{u_{p^{\prime\prime}}}-x_{u_{q^{\prime\prime}}}<0$

and

$x_{w_{m^{*}-b+1}}-x_{w_{m^{*}-a-1}}<x_{w_{m^{*}-b+2}}-x_{u_{m^{*}-a-2}}<\cdots<x_{u_{p^{\prime\prime}+1}}-x_{u_{q^{\prime\prime}}}\leq
0$

###### Proof.

For $p^{\prime\prime}+q^{\prime\prime}=m^{\prime\prime}$ and $1\leq
q^{\prime\prime}-p^{\prime\prime}\leq 2$, by Lemma 2.5, we have
$x_{u_{p^{\prime\prime}-i}}-x_{u_{q^{\prime\prime}+i}}$ having the same sign
as $\sigma(T^{u_{p^{\prime\prime}}})-\sigma(T_{u_{q^{\prime\prime}}})$ for
$0\leq i\leq p^{\prime\prime}+b-m^{*}$ and
$x_{w_{p^{\prime\prime}-i+1}}-x_{w_{q^{\prime\prime}+i}}$ having the same sign
as $\sigma(T^{u_{p^{\prime\prime}}})-\sigma(T_{u_{q^{\prime\prime}}})$ or
$x_{w_{p^{\prime\prime}-i+1}}-x_{w_{q^{\prime\prime}+i}}=0$ for $0\leq i\leq
p^{\prime\prime}+b-m^{*}$.
From Eq. (5), we have
$\sigma(T^{u_{m*-b-1}})-\sigma(T^{u_{p^{\prime\prime}}})-(\sigma(T_{u_{m^{*}-a}})-\sigma(T_{u_{q^{\prime\prime}}}))=c(\sigma(T^{u_{p^{\prime\prime}}})-\sigma(T_{u_{q^{\prime\prime}}}))$
for some positive number c. By Fact 1, we have
$\sigma(T^{u_{a}})>\sigma(T_{u_{m^{*}-a}})$. Since $a+b<m^{*}-2$, we have

$\displaystyle\sigma(T^{u_{p^{\prime\prime}}})-\sigma(T_{u_{q^{\prime\prime}}})=\sigma(T^{u_{p^{\prime\prime}}})-\sigma(T^{u_{a}})+\sigma(T^{u_{a}})-\sigma(T_{u_{q^{\prime\prime}}})$
$\displaystyle>\sigma(T^{u_{p^{\prime\prime}}})-\sigma(T^{u_{m*-b-1}})+\sigma(T^{u_{a}})-\sigma(T_{u_{m^{*}-a}})-(\sigma(T_{u_{q^{\prime\prime}}})-\sigma(T_{u_{m^{*}-a}}))$
$\displaystyle>-(\sigma(T^{u_{m^{*}-b-1}})-\sigma(T^{u_{p^{\prime\prime}}})-(\sigma(T_{u_{m^{*}-a}}))-\sigma(T_{u_{q^{\prime\prime}}}))=-c(\sigma(T^{u_{p^{\prime\prime}}})-\sigma(T_{u_{q^{\prime\prime}}})).$

Thus, $\sigma(T^{u_{p^{\prime\prime}}})-\sigma(T_{u_{q^{\prime\prime}}})>0$.
By Corollary 2.2, we have
$x_{u_{p^{\prime\prime}-i}}-x_{u_{q^{\prime\prime}+i}}<x_{u_{p^{\prime\prime}-i+1}}-x_{u_{q^{\prime\prime}+i-1}}<0$
and
$x_{w_{p^{\prime\prime}-i+1}}-x_{w_{q^{\prime\prime}+i}}<x_{w_{p^{\prime\prime}-i+2}}-x_{w_{q^{\prime\prime}+i-1}}\leq
0$ for $0\leq i\leq p^{\prime\prime}+b-m^{*}$. ∎

######  Lemma 3.1.

Let $k$-uniform hypertree $T=C_{k}(m^{*},\Delta,a,b)$ with $m^{*}>a+b$. For
$a+2\leq b$, we have $\rho(T)<\rho(C_{k}(m^{*},\Delta,a+1,b-1)).$

###### Proof.

Let $T^{\prime}$ be the graph obtained from $T$ by moving all pendant edges at
${u_{m*-b}}$ from to ${u_{a+1}}$. Then $T^{\prime}\cong
C_{k}(m^{*},\Delta,a+1,b-1)$.

Let $r=d(u_{m^{*}-b},u_{a+1})$ and take $\mathbf{x}=\mathbf{x}(T)$,
$H=T\setminus\\{V^{\prime}(S_{m^{*}-b})\\}$

By Eq. (6), we have the following.

$\frac{1}{2}(\rho(T^{\prime})-\rho(T))=\sigma(V^{\prime}(S_{m^{*}-b}))(W(H,u_{a+1})-W(H,u_{m^{*}-b})).$

To show $\rho(T^{\prime})>\rho(T)$, we only need to show
$J=W(H,u_{a+1})-W(H,u_{m^{*}-b})>0$.

Let
$c^{\prime}(r)=W_{0}(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{p^{\prime}+1})+r|V(T^{u_{a+1}})|$.
By Fact 3, $x_{u_{a+1}}>x_{u_{m^{*}-b}}$.

To show $J>0$, we only need to show the followings:

(1). $\rho(T)>c^{\prime}(r)$;

(2). $2J>(\rho(T)-c^{\prime}(r))(x_{u_{a+1}}-x_{u_{m^{*}-b}})$.

For (1), since $\rho(T)$ is bounded below the minimum sum of the row of
$D(T)$, it suffices to prove that the sum of each row of $D(T)$ is greater
than $c^{\prime}(r)$.

For $0\leq j\leq a$,

$\displaystyle W_{0}(T,u_{j})$ $\displaystyle\geq
W_{0}(T_{u_{a+1}},u_{a})+W_{0}(T^{u_{a}},u_{j})+W_{0}(V^{\prime}(T_{u_{a}}^{u_{a+1}}),u_{a})$
$\displaystyle>W_{0}(T_{m^{*}-a-1},u_{a})+W_{0}(T_{u_{a}}^{u_{m^{*}-b}},u_{a})>c^{\prime}(r).$

Then, $W_{0}(T,w_{j})=W_{0}(T,u_{j})+|V(T_{u_{j}})|-1>c^{\prime}(r)$.

For pendant vertices $v_{j}$ of $u_{j}$,
$W_{0}(T,v_{j})-W_{0}(T,u_{j})=|V(T)|-k>0$.

For $m^{*}-b\leq j\leq m^{*},u_{j},v_{j},w_{j}$ it can be proved by a similar
argument.

For $a+1\leq j\leq m^{*}-b-1$, we have
$W_{0}(V^{\prime}(T^{m^{*}-b}_{u_{a+1}}),u_{j})\geq
W_{0}(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{p^{\prime}+1})$, then

$\displaystyle W_{0}(T,u_{j})$
$\displaystyle=W_{0}(T^{u_{a+1}},u_{j})+W_{0}({T_{u_{m^{*}-b}}},u_{j})+W_{0}(V^{\prime}(T^{m^{*}-b}_{u_{a+1}}),u_{j})$
$\displaystyle>W_{0}(T^{u_{a+1}},u_{j})+W_{0}({T_{u_{m^{*}-a-1}}},u_{j})+W_{0}(V^{\prime}(T^{m^{*}-b}_{u_{a+1}}),u_{j})>c^{\prime}(r)$

Then, $W_{0}(T,w_{j})=W_{0}(T,u_{j})+|V(T_{u_{j}})|-1>c^{\prime}(r)$. Thus, we
have $\rho(T)>c^{\prime}(r)$.

For (2), by Eq. (7) and Fact 3, we have:

$\rho(T)(x_{u_{a+1}}-x_{u_{m^{*}-b}})=J+r\sigma(V^{\prime}(S_{m^{*}-b}))<2J+r\sigma(V^{\prime}(S_{m^{*}-a-1}))-J.$

we only need to show that:
$J>-c^{\prime}(r)(x_{u_{a+1}}-x_{m^{*}-b})+r\sigma(V^{\prime}(S_{m^{*}-a-1})).$

From $V(H)=V(T^{u_{a+1}})\sqcup V(T_{u_{m^{*}-a-1}})\sqcup
V^{\prime}(T_{u_{m^{*}-a-1}},u_{m^{*}-b})\sqcup
V^{\prime}(T_{m^{*}-b}^{m^{*}-a-1})\sqcup\\{u_{m^{*}-b}\\}$, we know that
$J=J_{1}+J_{2}+J_{3}+rx_{u_{m^{*}-b}}$, where

$\displaystyle J_{1}=$ $\displaystyle W(T^{u_{a+1}}\cup
T_{u_{m^{*}-a-1}},u_{a+1})-W(T^{u_{a+1}}\cup T_{u_{m^{*}-a-1}},u_{m^{*}-b}),$
$\displaystyle J_{2}=$ $\displaystyle
W(V^{\prime}(T_{m^{*}-b}^{m^{*}-a-1}),u_{a+1})-W(V^{\prime}(T_{m^{*}-b}^{m^{*}-a-1}),u_{m^{*}-b}),$
$\displaystyle J_{3}=$ $\displaystyle
W(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{a+1})-W(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{m^{*}-b}).$

Since $\forall w\in V^{\prime}(T_{m^{*}-b}^{m^{*}-a-1})$,
$d_{T}({u_{a+1}},w)-d_{T}(u_{m^{*}-b},w)>0$, we have $J_{2}>0.$

By Fact 1 and Fact
3,$x_{u_{a+1}}-x_{u_{m^{*}-b}}>x_{u_{a+1}}-x_{u_{m^{*}-a-1}}>0$,
note that
$-r|V(T^{u_{a+1}})|=W_{0}(T^{u_{a+1}},u_{a+1})-W_{0}(T^{u_{a+1}},u_{m^{*}-b})<0$,
we have

$\displaystyle J_{1}=W(T^{u_{a+1}}\cup
T_{u_{m^{*}-a-1}},u_{a+1})-W(T^{u_{a+1}}\cup T_{u_{m^{*}-a-1}},u_{m^{*}-b})$
$\displaystyle>$
$\displaystyle(W_{0}(T^{u_{a+1}},u_{a+1})-W_{0}(T^{u_{a+1}},u_{m^{*}-b}))(x_{u_{a+1}}-x_{u_{m^{*}-a-1}})+r\sigma(V^{\prime}(S_{m^{*}-a-1})$
$\displaystyle>$
$\displaystyle-r|V(T^{u_{a+1}})|(x_{u_{a+1}}-x_{u_{m^{*}-b}})+r\sigma(V^{\prime}(S_{m^{*}-a-1}),$

Note that
$-(W_{0}(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{p^{\prime}+1}))<W_{0}(V^{\prime}(T_{u_{a+1}}^{p^{\prime}+1}),u_{a+1})-W_{0}(V^{\prime}(T_{u_{a+1}}^{p^{\prime}+1}),u_{m^{*}-b})<0$,
by Fact 2, $x_{u_{a+1}}-x_{u_{m^{*}-b}}>0$, we have :

$\displaystyle
J_{3}=W(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{a+1})-W(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{m^{*}-b})$
$\displaystyle>(W_{0}(V^{\prime}(T_{u_{a+1}}^{p^{\prime}+1}),u_{a+1})-W_{0}(V^{\prime}(T_{u_{a+1}}^{p^{\prime}+1}),u_{m^{*}-b}))(x_{u_{a+1}}-x_{u_{m^{*}-b}})$
$\displaystyle>-(W_{0}(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{p^{\prime}+1}))(x_{u_{a+1}}-x_{u_{m^{*}-b}})$

Then

$\displaystyle J$
$\displaystyle=J_{1}+J_{2}+J_{3}+rx_{m^{*}-b}>-(r|V(T^{u_{a+1}})|+W_{0}(V^{\prime}(T_{u_{a+1}}^{m^{*}-b}),u_{p^{\prime}+1}))(x_{u_{a+1}}-x_{u_{m^{*}-b}})$
$\displaystyle+r\sigma(V^{\prime}(S_{m^{*}-a-1})+rx_{m^{*}-b}$
$\displaystyle>-c^{\prime}(r)(x_{u_{a+1}}-x_{m^{*}-b})+r\sigma(V^{\prime}(S_{m^{*}-a-1})).$

∎

######  Lemma 3.2.

Let $T$ be the $k$-uniform hypertree in $\mathbb{T}_{k}(m,\Delta,n)$ with
maximum distance spectral radius, where $\Delta\geq 3$ and $n\geq 2$. Then
each edge $e$ of $T$ has at most $2$ vertices in $e$ with degree at least $2$.

###### Proof.

If there is an edge $e$ of $T$, such that there are at least $3$ vertices of
degree at least $2$ in $e$. Let
$P=(u_{0},e_{1},u_{1},\cdots,u_{d-1},e_{d},u_{d})$ be one of the longest paths
in $T$ which contains $e$. Then there exists some vertex $w$ in $e$ such that
$E^{\prime}=E_{T}(w)-E(P)\neq\emptyset$. Suppose $u_{i},u_{i+1}\in e$,
$d_{P}(u_{i})=d_{P}(u_{i+1})=2$. Let $T_{1}$ be the component of $T-e$
containing $u_{0}$, $T_{2}$ be the component of $T-e$ containing $u_{d}$,
$H_{w}$ be the connected components of $T-e$ containing $w$, $T_{3}$ be the
component of $T\setminus(\\{u_{i}\\}\cup\\{u_{i+1}\\})$ containing $w$.
Suppose without loss of generality that $\sigma(T_{1})\geq\sigma(T_{2})$, then
let $T^{\prime}$ be the graph obtained from $T$ moving edges in $E^{\prime}$
from $w$ to a pendent neighbor $u_{d}$. Then
$T^{\prime}\in\mathbb{T}_{k}(m,\Delta,n)$. Note that:

$\displaystyle\frac{1}{2}$
$\displaystyle(\rho(T^{\prime})-\rho(T))\geq\frac{1}{2}\mathbf{x}^{T}(D(T^{\prime})-D(T))\mathbf{x}$
(18)
$\displaystyle>(d_{T}(u_{i+1},u_{d}))(\sigma(T_{1})-\sigma(T_{2}))\sigma(V^{\prime}(H_{w}))+((d_{T}(u_{i+1},u_{d})+1)x_{w}+x_{u_{d-1}}-x_{u_{d}})\sigma(V^{\prime}(H_{w})).$

And

$\rho(T)(x_{u_{d}}-x_{u_{d-1}})=\sigma(T_{1})+\sigma(T_{2})-\sigma(e_{d})+\sigma(T_{3})+x_{u_{d-1}}-x_{u_{d}}$
$\rho(T)x_{w_{i_{0}}}>\sigma(T_{1})+\sigma(T_{2})+\sigma(T_{3})-x_{w_{i_{0}j}}.$

Thus

$\rho(T)((d_{T}(u_{i+1},u_{d})+1)x_{w_{i_{0}}}+x_{u_{d-1}}-x_{u_{d}})>\sigma(e_{d})+x_{u_{d}}-x_{u_{d-1}}>0,$

we get $\rho(T^{\prime})>\rho(T)$ from (18), a contradiction. Thus each edge
$e$ of $T$ has at most $2$ vertices in $e$ with degree at least $2$. ∎

By a similar argument we can obtain the following conclusion:

######  Corollary 3.1.

Let $T$ be a caterpillar in $\mathbb{T}_{k}(m,\Delta,n)$, where $\Delta\geq 3$
and $n\geq 2$. Then each edge $e$ of $T$ has at most $2$ vertices in $e$ with
degree at least $2$.

Now, we are ready to identify the structure of the caterpillar in
$\mathbb{T}_{k}(m,\Delta,n)$ with maximum distance spectral radius. In graph
theory, a branch at a vertex $u$ in a tree is a maximal subtree containing $u$
as a pendent vertex.

######  Theorem 3.1.

Let $T$ be a caterpillar in $\mathbb{T}_{k}(m,\Delta,n)$ with $\Delta\geq 3$
and $n\geq 2$. Let $m^{*}=m-n(\Delta-2)$, then
$\rho(T)\leq\rho(C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil))$,
where equality holds if and only if $T\cong
C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil)$.

###### Proof.

Assume $T$ to be a caterpillar with the maximum distance spectral radius in
$\mathbb{T}_{k}(m,\Delta,n)$, and let $\mathbf{x}=\mathbf{x}(T)$. Suppose that
$P=(u_{0},e_{1},u_{1},\cdots,u_{d-1},e_{d},u_{d})$ is a diameter path of $T$.
By Corollary 3.1, for $w\in
P\setminus(\displaystyle\cup_{i=1}^{d-1}\\{u_{i}\\})$, $d_{T}(w)=1$.

Let $W$ be the set of vertices of degree $\Delta$ in $T$. Suppose that
$d_{T}(u_{i})\geq 3$ for some $u_{i}\in V(T)\setminus W$. Then there exists a
branch $T_{i}$ of $T$ at $u_{i}$ containing an edge $e\in
E_{T}(v_{i})\backslash E(P)$. Suppose without loss of generality that
$\sigma(T^{u_{i}})\geq\sigma(T_{u_{i}})$. Let $T^{\prime}$ be the graph
obtained from $T$ by moving edge $e_{i}$ from $u_{i}$ to $u_{d}$, then
$T^{\prime}$ is a caterpillar in $\mathbb{T}_{k}(m,\Delta,n)$. By Lemma 2.4
$\rho(T^{\prime})>\rho(T)$, a contradiction. Thus, the degree of any vertex of
$V(T)\setminus W$ is at most two.

If $H$ has no vertex of degree $2$, then we have $T\cong
C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil)$.
Otherwise, suppose that $d_{T}(u_{1})=d_{T}(u_{d-1})=2$ and
$d_{T}(u_{j})=\Delta$ for some $j$ with $2\leq j\leq d-2$. Suppose without
loss of generality that $\sigma(T^{u_{j}})\geq\sigma(T_{u_{j}})$. Let
$T^{\prime\prime}$ be the graph obtained from $T$ by moving the pendant edges
at $u_{j}$ from $u_{j}$ to $u_{d-1}$, then $T^{\prime\prime}$ is a caterpillar
in $\mathbb{T}_{k}(m,\Delta,n)$. By Lemma 2.4,
$\rho(T)<\rho(T^{\prime\prime})$, a contradiction. Thus,
$\max(d_{T}(u_{2}),d_{T}(u_{d-1}))=\Delta$. Without loss of generality,
suppose $d_{T}(u_{d-1})=\Delta$.

Assume that $d_{u_{1}}=2$ and let $U$ be the sets of vertices of degree $2$ in
$T$. If $T\setminus U$ has exactly one component with size greater than $1$,
then $T\cong C_{k}(m,{\Delta},0,n)$. By Lemma 3.1, we have
$\rho(T)=\rho(C_{k}(m^{*},{\Delta},0,n))<\rho(C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil))$,
a contradiction. Therefore, $T\setminus U$ has at least two non-trivial
components and there are vertices $u_{i}$ and $u_{j}$ with $2\leq i<j\leq d-2$
such that $d_{T}(u_{i})=\Delta$ and $d_{T}(u_{j})=2$. If
$\sigma(T^{u_{i}})\geq\sigma(T_{u_{i}})$, let $T_{1}^{\prime}$ be the graph
obtained from $T$ by moving the pendant edges at $u_{i}$ from $u_{i}$ to
$u_{j}$. By Lemma 2.4 we have $\rho(T_{1}^{\prime})>\rho(T)$, a contradiction.
If $\sigma(T^{u_{i}})<\sigma(T_{u_{i}})$, let $T_{2}^{\prime}$ be the graph
obtained from $T$ by moving the pendant edges at $u_{i}$ from $u_{i}$ to
$u_{1}$, by Lemma 2.4 we have $\rho(T_{2}^{\prime})>\rho(T)$, a contradiction.
Then $d_{u_{1}}=\Delta$, and $T\setminus U$ has at least two components with
size greater than $1$.

Suppose $T\setminus U$ has at least $3$ components with size greater than $1$.
Then there are three vertices $u_{i},u_{j},u_{l}$ with $2\leq i<j<l\leq d-2$
in $T$ such that $d_{T}(u_{j})=\Delta,d_{T}(i)=d_{T}(l)=2$. Suppose without
loss of generality that $\sigma(T^{u_{j}})\geq\sigma(T_{u_{j}})$. Let
$T^{\prime\prime\prime}$ be the graph obtained from $T$ by moving the pendant
edges at $u_{j}$ from $u_{j}$ to $u_{l}$, then $T^{\prime\prime\prime}$ is a
caterpillar in $\mathbb{T}_{k}(m,\Delta,n)$. By Lemma 2.4
$\rho(T^{\prime\prime\prime})>\rho(T)$, a contradiction.Thus $T\setminus U$
has exactly $2$ components with size greater than $1$, then $T\cong
C_{k}(m^{*},\Delta,a,b)$, for some $a+b=n,a\geq 1,b\geq 1$. By Lemma 3.1,
$T\cong
C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil).$ ∎

##  4 Distance spectral radius of trees in $\mathbb{T}_{k}(m,\Delta,n)$

In the following section, we turn our attention to finding the extreme graph
within the class $\mathbb{T}_{k}(m,\Delta,n)$ that maximizes the distance
spectral radius. We begin by considering the graph $H=G_{c}(s,t)$. Set
$\mathbf{x}=\mathbf{x}(H)$, following the notation used in Lemma 2.3, we
denote $x_{v_{i}}$ for the component of $\mathbf{x}$ pertaining to the pendant
neighbor of $u_{i}$ for $i\neq s$, set one of these vertices as $v_{i}$, and
let $x_{w_{i}}$ be the component of $\mathbf{x}$ that corresponds to the
vertices in $e_{i}$ of degree 1.

For $i=1,\cdots,c$, let $h_{i},f_{i}$ be two vertices of a pendant edge
$e_{s,i}$ of $k$-uniform hypergraph $G_{i}$ with $d_{G_{i}}(h_{i})=1$ and
$d_{G_{i}}(f_{i})>1$ if $|E(G_{i})|>1$. $F_{i}=\\{f_{i}\\}$ for $|H_{i}|=1$,
and $F_{i}$ is the non-trivial connected component of $H_{i}-e_{s,i}$ for
$|E(H_{i})|>1$. The rooted graph $G_{r_{s}}$ of $G_{c}(s,t)$ is the graph
obtained from $G_{1},\ldots,G_{c}$ identifying $h_{1},\ldots,h_{c}$ as a new
vertex $r_{s}$, which is labeled as the root of $G_{r_{s}}$. For the graph
$H=G_{c}(s,t)$, set $\mathbf{x}=\mathbf{x}(H)$, let $q_{i}$ be one of the
vertices of degree one in $e_{s,i}\setminus({h_{i}}\cup{f_{i}})$. By Lemma
2.3, we denote $x_{q_{i}}$ the component of $\mathbf{x}$ that corresponds to
any vertex of degree one in $e_{s,i}\setminus(\\{h_{i}\\}\cup\\{f_{i}\\})$. We
have following lemma holds:

######  Lemma 4.1.

Setting $|E(G_{1})|>1$, for $s\geq t\geq 2$, we have

$\rho(G_{c}(s+1,t-1))>\rho(G_{c}(s,t))$

holds if one of the following conditions satisfied: (i) $c=1$; (ii) $c\geq 2$,
$G_{1}$ is a loose path and $|E(G_{i})|=1$ for $2\leq i\leq c$; (iii) $c\geq
2$ and there is a vertex of $V(G_{1})$ with at least $c$ adjacent pendant
edges in $G_{c}(s,t)$.

###### Proof.

Take $H=G_{c}(s,t)$, $\mathbf{x}=\mathbf{x}(H)$. For $(i)$, Let $H^{\prime}$
be the graph obtained from $H$ by moving $E_{G_{u_{s}}}(u_{s})$ from $u_{s}$
to $u_{s+1}$, moving $E_{G_{u_{s+1}}}(u_{s+1})$ from $u_{s+1}$ to $u_{s}$,
then $H^{\prime}\cong G_{1}(s+1,t-1)$. Using Eq. (8), we have

$\displaystyle\frac{1}{2}(\rho(H^{\prime})-\rho(H))$
$\displaystyle\geq\frac{1}{2}\mathbf{x}^{T}(D(H^{\prime})-D(H))\mathbf{x}$
$\displaystyle=(\sigma(V^{\prime}(G_{u_{s}}))-\sigma(V^{\prime}(G_{u_{s+1}}))(\sigma(V^{\prime}(H^{u_{s}}))+x_{u_{s}}-\sigma(V^{\prime}(H_{u_{s+1}}))-x_{u_{s+1}}).$

By Lemma 2.6, we have
$\sigma(V^{\prime}(H^{u_{s}}))+x_{u_{s}}>\sigma(V^{\prime}(H_{u_{s+1}}))+x_{u_{s+1}}$.
Hence, to prove $\rho(G^{\prime})>\rho(G)$, it suffices to show that
$\sigma(V^{\prime}(G_{u_{s}}))-\sigma(V^{\prime}(G_{u_{s+1}}))>0$.

Choose $z\in V(G_{u_{s}})$ such that $d_{H}(z,f_{1})=\max_{v\in
V(F_{1})}d_{H}(v,f_{1})=d$, then $z$ is in a pendant edge $e^{\prime}$, let
the set of pendant vertices in $e^{\prime}$ be $Z$. From the eigenequation of
$G$ we have

$\displaystyle\rho(G)x_{f_{1}}=d(k-1)x_{z}+3(k-1)x_{v_{s+1}}+x_{u_{s}}+W(V^{\prime}(G_{u_{s}}\setminus
Z),f_{1})+W(H\setminus(V(G_{u_{s}})\cup V^{\prime}(G_{u_{s+1}})),f_{1}),$
$\displaystyle\rho(G)x_{q_{1}}\geq(d+1)(k-1)x_{z}+3(k-1)x_{v_{s+1}}-x_{q_{1}}+x_{u_{s}}$
$\displaystyle\ \ +W(V^{\prime}(G_{u_{s}}\setminus
Z),f_{1})+W(G\setminus(V(G_{u_{s}})\cup V^{\prime}(G_{u_{s+1}})),q_{1}),$
$\displaystyle\rho(G)x_{z}=(k-2)x_{z}+(d+3)(k-1)x_{v_{s+1}}+(d+1)x_{u_{s}}+W(V^{\prime}(G_{u_{s}}\setminus
Z),z)+W(H\setminus(V(G_{u_{s}})\cup V^{\prime}(G_{u_{s+1}})),z),$
$\displaystyle\rho(G)x_{v_{s+1}}\leq(d+3)(k-1)x_{z}+(k-2)x_{v_{s+1}}+2x_{u_{s}}+W(V^{\prime}(G_{u_{s}}\setminus
Z),f_{1})$ $\displaystyle\ \ +3\sigma(V^{\prime}(G_{u_{s}}\setminus
Z))+W(G\setminus(V(G_{u_{s}})\cup V^{\prime}(G_{u_{s+1}})),v_{s+1})$

Note that for $w\in V(H)\setminus(V(G_{u_{s}})\cup V^{\prime}(G_{u_{s+1}}))$,

$d_{H}(f_{1},w)+(k-2)d_{H}(q_{1},w)+(k-1)d_{H}(z,w)-(k-1)d_{H}(v_{s+1},w)\geq
0,$

we have

$\displaystyle\rho(G)(x_{f_{1}}+(k-2)x_{q_{1}}+(k-1)x_{z}-(k-1)x_{v_{s+1}})$
$\displaystyle\geq$ $\displaystyle(1-k^{2})x_{z}+(d+3)(k-1)^{2}x_{v_{s+1}}$
$\displaystyle-$
$\displaystyle(k-2)x_{q_{1}}+(k-1)(W(V^{\prime}(G_{u_{s}}\setminus
Z),z)-3\sigma(V^{\prime}(G_{u_{s}}\setminus Z))).$

Then by $\sigma(V^{\prime}(G_{u_{s}}))=\sigma(V^{\prime}(G_{u_{s}}\setminus
Z))+(k-1)x_{z}$ and $\sigma(V^{\prime}(G_{u_{s}}))>x_{q_{1}}$, we have:

$\displaystyle(\rho(G)+4(k-1))(\sigma(V^{\prime}(G_{s}))-\sigma(V^{\prime}(G_{s+1})))$
$\displaystyle\geq\rho(G)(x_{f_{1}}+(k-2)x_{q_{1}}+(k-1)(x_{z}-x_{v_{s+1}}))+4(k-1)(\sigma(V^{\prime}(G_{s}))-\sigma(V^{\prime}(G_{s+1})))$
$\displaystyle>\rho(G)(x_{f_{1}}+(k-2)x_{q_{1}}+(k-1)(x_{z}-x_{v_{s+1}}))+3(k-1)(\sigma(V^{\prime}(G_{s}))+(k-1)x_{q_{1}}-4(k-1)\sigma(V^{\prime}(G_{s+1}))$
$\displaystyle>(d-1)(k-1)^{2}x_{v_{s+1}}+(k-1)(W(V^{\prime}(G_{u_{s}}\setminus
Z),z))>0.$

Thus $\sigma(V^{\prime}(G_{s}))-\sigma(V^{\prime}(G_{s+1}))>0$.

For $(ii)$, let $e$ be the adjacent edge of $f_{1}$ different from $e_{1}$,
$H^{\prime\prime}$ be the graph obtained from $H$ by moving edge $e$ from
$f_{1}$ to $v_{s+1}$, then $H^{\prime\prime}\cong G_{c}(s+1,t-1)$. Then

$\displaystyle\frac{1}{2}(\rho(H^{\prime\prime})-\rho(H))$
$\displaystyle\geq\frac{1}{2}\mathbf{x}^{T}(D(H^{\prime\prime})-D(H))\mathbf{x}$
(19)
$\displaystyle=(\sigma(F_{1})-x_{f_{1}})(\sigma(V^{\prime}(H^{u_{s}}))+x_{u_{s}}+3x_{f_{1}}+2(k-1)x_{q_{1}}+(c-1)(k-1)x_{q_{2}})$
$\displaystyle-(\sigma(F_{1})-x_{f_{1}})(\sigma(V^{\prime}(H_{u_{s}}))-(k-2)x_{w_{s+1}}+kx_{v_{s+1}})$
$\displaystyle>(\sigma(F_{1})-x_{f_{1}})(x_{u_{s}}+3x_{f_{1}}+2(k-2)x_{q_{1}}+(k-1)x_{q_{2}}-kx_{v_{s+1}})$

For any $w\in V(G)\setminus(V(e_{s,1})\cup V(e_{s,2})\cup V(e_{s+1,1}))$, we
have

$d_{H}(w,u_{s})+3d_{H}(w,f_{1})+2(k-2)d_{H}(q_{1},w)+(k-1)d_{H}(q_{2},w)-kd_{H}(v_{s+1},w)\geq
0$

Thus

$\displaystyle\rho(G)(x_{u_{s}}+3x_{f_{1}}+2(k-2)x_{q_{1}}+(k-1)x_{q_{2}}-kx_{v_{s+1}})$
$\displaystyle\geq(k-2)x_{u_{s}}+(k-5)x_{f_{1}}+(k-2)(k-4)x_{q_{1}}+(2k-3)(k-1)x_{q_{2}}+(8k^{2}-11k+4)x_{v_{s+1}},$

and then

$\displaystyle(\rho(G)+1)(x_{u_{s}}+3x_{f_{1}}+2(k-2)x_{q_{1}}+(k-1)x_{q_{2}}-kx_{v_{s+1}})$
$\displaystyle\geq(k-1)x_{u_{s}}+(k-2)x_{f_{1}}+(k-2)^{2}x_{q_{1}}+2(k-1)^{2}x_{q_{2}}+4(2k-1)(k-1)x_{v_{s+1}}>0.$

Therefore,
$x_{u_{s}}+3x_{f_{1}}+2(k-2)x_{q_{1}}+(k-1)x_{q_{2}}-kx_{v_{s+1}}>0$, from
(19), we have $\rho(G^{\prime\prime})>\rho(G)$.

For $(iii)$, let $y\in G_{1}$ such that $Y$ is a hyperstar $S^{k}_{c}$ which
edges in $E(Y)$ are pendant edge of $y$. Denote $V^{\prime}(Y)=V(Y)-\\{y\\}$,
by Lemma 2.3, we denote $x_{y^{\prime}}$ for the component of $\mathbf{x}$
pertaining to the vertices in $V^{\prime}(Y)$. Let $H^{\prime\prime\prime}$ be
the graph obtained from $H$ by moving $E_{G_{u_{s}}}(u_{s})$ from $u_{s}$ to
$u_{s+1}$, moving $E_{G_{u_{s+1}}}(u_{s+1})$ from $u_{s+1}$ to $u_{s}$, then
$H^{\prime\prime\prime}\cong G_{c}(s+1,t-1)$. Then by Eq. (8)

$\displaystyle\frac{1}{2}(\rho(H^{\prime\prime\prime})-\rho(H))$
$\displaystyle\geq\frac{1}{2}\mathbf{x}^{T}(D(H^{\prime\prime\prime})-D(H))\mathbf{x}$
(20)
$\displaystyle=(\sigma(V^{\prime}(G_{u_{s}}))-\sigma(V^{\prime}(G_{u_{s+1}})))(\sigma(V^{\prime}(H^{u_{s}}))+x_{u_{s}}-\sigma(V^{\prime}(H_{u_{s+1}}))-x_{u_{s+1}}).$

By Lemma 2.6, we have
$\sigma(V^{\prime}(H^{u_{s}}))+x_{u_{s}}>\sigma(V^{\prime}(H_{u_{s+1}}))+x_{u_{s+1}}$.
To show $\rho(G^{\prime\prime\prime})>\rho(G)$, we only need to show that

$\sigma(V^{\prime}(G_{u_{s}}))\geq\sigma(V^{\prime}(G_{u_{s+1}})).$

Take

$\displaystyle V_{1}$
$\displaystyle=V^{\prime}(H^{u_{s}}),V_{2}=V^{\prime}(H_{u_{s+1}}),V_{3}=V(e_{s+1}),V_{4}=V(\cup_{j=2}^{c}F_{j}),$
$\displaystyle V_{5}$ $\displaystyle=V(F_{1}\setminus
V^{\prime}(Y)),V_{6}=N_{G_{u_{s}}}(u_{s})\setminus\cup_{j=1}^{c}\\{f_{i}\\},V_{7}=V^{\prime}(Y),V_{8}=V^{\prime}(G_{u_{s+1}})$

Let $I=N_{G_{u_{s}}}(u_{s})\cup V^{\prime}(Y)$. Note that for $v\in V_{5}\cup
V_{7}$, $d_{H}(v,q_{i})\geq d_{H}(v,f_{i})$ and $d_{H}(y,f_{1})\geq 0$, we
have:

$\displaystyle WD(V_{1};I,V^{\prime}(G_{u_{s+1}}))$
$\displaystyle=c(k-1)W(V^{\prime}(H^{u_{s}}),u_{s})+(d_{H}(y,f_{1})+1)\sigma(V^{\prime}(H^{u_{s}}))>0,$
$\displaystyle WD(V_{2};I,V^{\prime}(G_{u_{s+1}}))$
$\displaystyle=c(k-1)W(V^{\prime}(H_{u_{s+1}}),u_{s+1})+(d_{H}(y,f_{1})+4)\sigma(V^{\prime}(H_{u_{s+1}}))>0,$
$\displaystyle WD(V_{3};I,V^{\prime}(G_{u_{s+1}}))$
$\displaystyle=c(k-1)\big{(}(d_{H}(y,f_{1})+4)x_{u_{s+1}}+(k-2)(d_{H}(y,f_{1})+3)x_{w_{s+1}}+(1+d_{H}(y,f_{1}))x_{u_{s}}\big{)}>0,$
$\displaystyle WD(V_{4};I,V^{\prime}(G_{u_{s+1}}))$
$\displaystyle>WD(V_{4};V^{\prime}(Y),V^{\prime}(G_{u_{s+1}}))\geq 0,$
$\displaystyle WD(V_{5};I,V^{\prime}(G_{u_{s+1}}))$
$\displaystyle\geq\sum_{i=1}^{c}(k-1)W(V_{5},x_{f_{i}})+WD(V_{5};V^{\prime}(Y),V^{\prime}(G_{u_{s+1}}))$
$\displaystyle=(k-1)\sum_{i=1}^{c}\big{(}W(F_{1}\setminus
V^{\prime}(Y),y)+(d_{H}(f_{1},f_{i})-2)\sigma(F_{1}\setminus
V^{\prime}(Y))\big{)}>-2c(k-1)\sigma(F_{1}\setminus V^{\prime}(Y)),$
$\displaystyle WD(V_{6};I,V^{\prime}(G_{u_{s+1}}))$
$\displaystyle\geq\sum_{i=1}^{c}(k-2)(ck-k-c)x_{q_{i}}\geq 0,$ $\displaystyle
WD(V_{7};I,V^{\prime}(G_{u_{s+1}}))$
$\displaystyle\geq\sum_{i=1}^{c}(k-1)W(V_{7},x_{f_{i}})+WD(V_{7};V^{\prime}(Y),V^{\prime}(G_{u_{s+1}}))$
$\displaystyle\geq(k-1)c\big{(}(k-1)c+(2-3k)\big{)}x_{y^{\prime}},$
$\displaystyle WD(V_{8};I,V^{\prime}(G_{u_{s+1}}))$
$\displaystyle\geq(5ck-5c+k)\sigma(V^{\prime}(G_{u_{s+1}}))>5c(k-1)\sigma(V^{\prime}(G_{u_{s+1}})).$

By $I\subseteq V^{\prime}(G_{u_{s}})$, and $V(G)=\bigsqcup_{i=1}^{8}V_{i}$, we
have :

$\displaystyle\rho(G)(\sigma(I)-\sigma(V^{\prime}(G_{u_{s+1}})))=\sum_{i=1}^{8}WD(V_{i};I,V^{\prime}(G_{u_{s+1}}))$
(21) $\displaystyle>c(k-1)\big{(}-2\sigma(F_{1}\setminus
V^{\prime}(Y))+((c-3)k-c+2)x_{y^{\prime}}+5\sigma(V^{\prime}(G_{u_{s+1}}))\big{)}.$

From $\sigma(V^{\prime}(G_{u_{s}}))-\sigma(F_{1}\setminus
V^{\prime}(Y))>\sigma(V^{\prime}(Y))$, we have

$\displaystyle(\rho(G)+2c(k-1))(\sigma(V^{\prime}(G_{u_{s}}))-V^{\prime}(G_{u_{s+1}}))$
$\displaystyle>c(k-1)\big{(}2\sigma(V^{\prime}(Y))+((k-1)c+(2-3k))x_{y^{\prime}}+3\sigma(V^{\prime}(G_{u_{s+1}}))\big{)}$
$\displaystyle>c(k-1)\big{(}(3(k-1)(c-1)-1)x_{y^{\prime}}+3\sigma(V^{\prime}(G_{u_{s+1}}))\big{)}$
$\displaystyle>3c(k-1)\sigma(V^{\prime}(G_{u_{s+1}}))>0$

which implies that
$\sigma(V^{\prime}(G_{u_{s}}))>\sigma(V^{\prime}(G_{u_{s+1}}))$. Thus,
$\rho(G^{\prime\prime\prime})>\rho(G)$. ∎

Now, we are ready to identify the structure of the hypertree in
$\mathbb{T}_{k}(m,\Delta,n)$ with maximum distance spectral radius.

######  Theorem 4.1.

Let $T$ be the $k$-uniform hypertree in $\mathbb{T}_{k}(m,\Delta,n)$ with
maximum distance spectral radius, where $\Delta\geq 3$ and $n\geq 2$, then
$T\cong
C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil)$, where
$m^{*}=m-n(\Delta-2)$.

###### Proof.

Let $\mathbf{x}=\mathbf{x}(T)$. By Lemma 3.2,each edge $e$ of $T$ has at most
$2$ vertices in $e$ with degree at least $2$, then by an argument analogous to
the one given in the proof of Theorem 3.1, the degree of any vertex of degree
less than $\Delta$ is at most $2$.

Let $U$ be the set of vertices of degree $2$ in $T$ and $|U|=k_{2}$. For any
$u\in V(T)$ with degree $\Delta$, let
$E_{T}(u)=\\{e_{1},e_{2},\cdots,e_{\Delta}\\}$ and $T_{i}$ be the branch of
$T$ at $u$ containing $e_{i}$, for $1\leq i\leq\Delta$. Now, we prove that
$U\subseteq V(T_{i})$ for some $i$ with $1\leq i\leq\Delta$. Suppose for the
sake of contradiction that $w_{1},w_{2}\in U,\ w_{1}\in V(T_{1})$, and
$w_{2}\in V(T_{2})$. Without loss of generality, assume that
$\sigma(T_{1})\geq\sigma(T_{2})$. Let $T^{\prime\prime}$ be the graph obtained
from $T$ by moving the edges $e_{3},\cdots e_{\Delta}$ from $u$ to $w_{2}$.
Then $T^{\prime\prime}\in\mathbb{T}_{k}(m,\Delta,n)$, by Lemma 2.4,
$\rho(T^{\prime\prime})>\rho(T)$, a contradiction. Then we have $U\subseteq
V(T_{i})$ for some $i$ with $1\leq i\leq\Delta$.

Suppose that $T$ is not a caterpillar, we have $T\cong G_{\Delta-2}(s,t)$ for
some nontrivial hypergraph $G_{r_{s}}$, and some integers $s$ and $t$ with
$s\geq t\geq 2$. Obviously,
$G_{\Delta-2}(s+1,t-1)\in\mathbb{T}_{k}(m,\Delta,n)$. If $\Delta=3$, then by
Lemma 4.1 $(i)$, $\rho(G_{\Delta-2})(s+1,t-1)>\rho(T)$, a contradiction.

Suppose that $\Delta\geq 4$. If $m=n(\Delta-1)+1$, then $U=\emptyset$,
implying that $V^{\prime}(G_{r_{s}})$ contains a vertex $v$ with degree
$\Delta$, then $v$ has at least $\Delta-2$ adjacent pendant edges in $T$. By
Lemma 4.1 $(iii)$, $\rho(G_{\Delta-2}(s+1,t-1))>\rho(T)$, a contradiction.
Suppose that $m>n(\Delta-1)+1$, then $k_{2}\geq 1$, if $V^{\prime}(G_{r_{s}})$
contains a vertex of degree $\Delta$, then there is a vertex in
$V^{\prime}(G_{r_{s}})$ with at least $\Delta-2$ adjacent pendant edges in
$T$, according to Lemma 4.1 $(iii)$, $\rho(G_{\Delta-2}(s+1,t-1))>\rho(T)$, a
contradiction. Therefore, $V^{\prime}(G_{r_{s}})$ contains no vertex of degree
$\Delta$ in $T$, that is, each vertex of $V(G_{r_{s}})$ is of degree $1$ or
$2$. Since $T$ is not a caterpillar, we have $G_{i_{0}}$ is a loose path for
some $1\leq i_{0}\leq c$ and $E(G_{i})=1$ for $i\neq i_{0}$, according to
Lemma 4.1 $(ii)$, $\rho(G_{\Delta-2}(s+1,t-1))>\rho(T)$, a contradiction. Thus
$T$ is a caterpillar in $\mathbb{T}_{k}(m,\Delta,n)$, then by Theorem 3.1,
$T\cong
C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil).$ ∎

We have shown that if $T$ is a $k$-uniform hypertree in
$\mathbb{T}_{k}(m,\Delta,n)$ with maximum distance spectral radius, then it
must be isomorphic to
$C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil)$ where
$m^{*}=m-n(\Delta-2)$. Furthermore, if $T$ is a $k$-uniform hypertree with $m$
edges and $n$ vertices of maximum degree having maximum distance spectral
radius, where $n\geq 2$. Take
$H_{1}=C_{k}(m_{1},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil)$,
$H_{2}=C_{k}(m_{2},\Delta-1,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil)$,
where $|E(H_{1})|=|E(H_{2})|=m$, $\Delta\leq\frac{m-1}{n}$, then by similar
argument as in the proof of Theorem 4.1 and using Lemma 2.4, we have:

$\rho(H_{1})<\rho(H_{2}),$

we can obtain the following corollary:

######  Corollary 4.1.

If $T$ is a $k$-uniform hypertree with $m$ edges and $n$ vertices of maximum
degree having maximum distance spectral radius, where $2\leq
n\leq\lfloor\frac{m-1}{2}\rfloor$. Then
$\rho(T)\leq\rho(C_{k}(m^{*},3,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil))$,
with equality if and only if $T\cong
C_{k}(m^{*},3,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil)$, where
$m^{*}=m-n$.

We study the effects of the graft transformations in two specific cases on the
distance spectral radius of connected uniform hypergraphs. Based on Lemma 4.1,
we can make following conjecture.

######  Conjecture 4.1.

$|E(G_{r_{s}})|>c$ and $d_{G_{r_{s}}}(r_{s})\geq c$, then for $s\geq t\geq 2$,

$\rho(G_{c}(s+1,t-1))>\rho(G_{c}(s,t)).$

About the graft transformation at different vertices, Lemma 3.1 presents a
special case, based on which we can propose the following question.

######  Question 1.

Given uniform hypergraphs $H$ and $G_{r}$ and the integers with $p\geq q\geq
1$, what conditions ensure that
$\rho(H_{u,v}(p,q;G_{r}))>\rho(H_{u,v}(p+1,q-1;G_{r}))$?

Let $P_{1}=(u_{0},e_{1},u_{1},e_{2},u_{2})$ with $|e_{1}|=9$, $|e_{2}|=3$,
$H=P_{2}=(u,e_{3},v)$ with $e_{3}=3$, $d_{H}(u)=d_{H}(v)=1$,
$P_{3}=(u_{3},e_{4},u_{4},e_{5},u_{5})$ with $|e_{4}|=3$, $|e_{5}|=6$. Let
$H^{\prime}$ be the hypergraph obtained from $H$ by attaching $u_{2}$, $u_{3}$
at $u$, $v$, respectively, and $H^{\prime\prime}$ be the hypergraph obtained
form $H^{\prime}$ by moving $e_{1}$ from $u_{1}$ to $u_{5}$. By calculation,
we have $\rho(H)\approx 53.04$, $\rho(H^{\prime})\approx 46.91$, then
$\rho(H)>\rho(H^{\prime})$, which shows that Lemma 2.2 does not hold in the
case of non-uniform hypergraphs.

For the $3$-uniform hypergraph $H=C_{3}(5,3,1,2)\in\mathbb{T}_{3}(8,3,3)$, let
$H^{\prime}$ be the non-uniform hypertree obtained from $H$ by moving a vertex
of degree $1$ in $e_{2}$ from $e_{2}$ to $e_{1}$. By calculation, we have
$\rho(H)\approx 45.33$, $\rho(H^{\prime})\approx 46.31$, then
$\rho(H^{\prime})>\rho(H)$. Hence,
$T=C_{k}(m^{*},\Delta,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil)$ is
not the hypertree with the maximal distance spectral radius among hypertrees
with fixed order $|V(T)|$, size $|E(T)|$, maximal degree $\Delta$, and number
of vertices of maximum degree. It is nature to ask the following question.

######  Question 2.

Which hypertrees have the the maximum distance spectral radius among all
hypertrees with fixed order, size, maximal degree, and number of vertices of
maximum degree?

[sorting=none]

## References

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## References

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  * [15] R.L Graham and L Lovász  “Distance matrix polynomials of trees”  In _Advances in Mathematics_ 29.1, 1978, pp. 60–88  DOI: https://doi.org/10.1016/0001-8708(78)90005-1
  * [16] Ronald L. Graham and Henry O. Pollak  “On the addressing problem for loop switching”  In _Bell System Technical Journal_ 50, 1971, pp. 2495–2519  URL: https://api.semanticscholar.org/CorpusID:115513657
  * [17] Hongying Lin and Bo Zhou  “Distance spectral radius of uniform hypergraphs”  In _Linear Algebra Appl._ 506, 2016, pp. 564–578  DOI: 10.1016/j.laa.2016.06.011
  * [18] Hongying Lin and Bo Zhou  “On distance spectral radius of uniform hypergraphs with cycles”  In _Discret. Appl. Math._ 239, 2018, pp. 125–143  URL: https://api.semanticscholar.org/CorpusID:4338662
  * [19] Hongying Lin and Bo Zhou  “The distance spectral radius of trees”  In _Linear and Multilinear Algebra_ 67, 2019, pp. 370–390  URL: https://api.semanticscholar.org/CorpusID:126394115
  * [20] Huiqiu Lin, Jinlong Shu, Jie Xue and Yuke Zhang  “A survey on distance spectra of graphs”  In _Adv. Math., Beijing_ 50.1, 2021, pp. 29–76
  * [21] Xiangxiang Liu, Ligong Wang and Xihe Li  “Distance spectral radii of $k$-uniform hypertrees with given parameters”  In _Linear Multilinear Algebra_ 69.14, 2021, pp. 2558–2571  DOI: 10.1080/03081087.2019.1682497
  * [22] Dragan Stevanović  “Walk counts and the spectral radius of graphs”  In _Bull. Cl. Sci. Math. Nat. Sci. Math._ , 2015, pp. 33–57
  * [23] Yanna Wang and Bo Zhou  “A proof of a conjecture on the distance spectral radius”  In _Linear Algebra Appl._ 674, 2023, pp. 124–154  DOI: 10.1016/j.laa.2023.05.025
  * [24] Yanna Wang and Bo Zhou  “Extremal properties of the distance spectral radius of hypergraphs”  In _Electron. J. Linear Algebra_ 36, 2020, pp. 411–429  URL: journals.uwyo.edu/index.php/ela/article/view/5121