††footnotetext: E-mails<EMAIL_ADDRESS>
# Walks, infinite series and spectral radius of graphs

Wenqian Zhang
School of Mathematics and Statistics, Shandong University of Technology
Zibo, Shandong 255000, P.R. China

###### Abstract

For a graph $G$, the spectral radius $\rho(G)$ of $G$ is the largest
eigenvalue of its adjacency matrix. In this paper, we seek the relationship
between $\rho(G)$ and the walks of the subgraphs of $G$. Especially, if $G$
contains a complete multi-partite graph as a spanning subgraph, we give a
formula for $\rho(G)$ by using an infinite series on walks of the subgraphs of
$G$. These results are useful for the current popular spectral extremal
problem.

Keywords: spectral radius; spectral extrema; walk; infinite series.
2020 Mathematics Subject Classification: 05C50.

##  1 Introduction

All graphs considered in this paper are finite, undirected and without
parallel edges or loops. For a graph $G$, let $\overline{G}$ be its
complement. The vertex set and edge set of $G$ are denoted by $V(G)$ and
$E(G)$, respectively. For a vertex $u$, let $d_{G}(u)$ be its degree. For two
vertices $u$ and $v$, we say $u\sim v$ if they are adjacent in $G$. For a
certain integer $n$, let $K_{n}$ and $K_{1,n-1}$ be the complete graph and the
star of order $n$, respectively. For $\ell\geq 2$ vertex-disjoint graphs
$G_{1},G_{2},...,G_{\ell}$, let $\cup_{1\leq i\leq\ell}G_{i}$ be the disjoint
union of them. Let $G_{1}\vee G_{2}$ be the join of $G_{1}$ and $G_{2}$. For
any terminology used but not defined here, one may refer to [2].

Let $G$ be a graph with vertices $v_{1},v_{2},...,v_{n}$. The adjacency matrix
of $G$ is an $n\times n$ matrix $(a_{ij})$, where $a_{ij}=1$ if $v_{i}\sim
v_{j}$, and $a_{ij}=0$ otherwise. The spectral radius $\rho(G)$ of $G$ is the
largest eigenvalue of its adjacency matrix. By Perron–Frobenius theorem,
$\rho(G)$ has a non-negative eigenvector. Any non-negative eigenvector
corresponding to $\rho(G)$ is called a Perron vector of $G$. If $G$ is
connected, any Perron vector of $G$ has positive entries.

For a family $\mathcal{H}$ of graphs, a graph $G$ is call $\mathcal{H}$-free
if $G$ does not contain any one in $\mathcal{H}$ as a subgraph. Let ${\rm
EX}(n,\mathcal{H})$ be the set of $\mathcal{H}$-free graphs of order $n$ with
the maximum number of edges, and let ${\rm SPEX}(n,\mathcal{H})$ be the set of
$\mathcal{H}$-free graphs of order $n$ with the maximum spectral radius. For
$r\geq 2,n\geq r$, let $T_{n,r}$ be the Turán graph of order $n$ with $r$
parts. Turán’s famous theorem states that ${\rm
EX}(n,\left\\{K_{r+1}\right\\})=\left\\{T_{n,r}\right\\}$. Nikiforov [17]
proves that ${\rm SPEX}(n,\left\\{K_{r+1}\right\\})=\left\\{T_{n,r}\right\\}$.
In 2010, Nikiforov [16] formally presented a spectral version of Turán-type
problem: what is the maximum spectral radius of an $\mathcal{H}$-free graph of
order $n$? In recent years, this problem has become very popular (for example,
see [1, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 18, 19]). In particular,
Wang, Kang and Xue [18] proved the following nice result, which is conjectured
by Cioabă, Desai and Tait [5].

###### Theorem 1.1

(Wang, Kang and Xue [18]) For $r\geq 2$ and sufficiently large $n$, let
$\mathcal{H}$ be a finite family of graphs such that all graphs in ${\rm
EX}(n,\mathcal{H})$ are obtained from $T_{n,r}$ by embedding $k$ edges into
its $r$ parts, where $k$ is a bounded integer. Then ${\rm
SPEX}(n,\mathcal{H})\subseteq{\rm EX}(n,\mathcal{H})$.

In general, ${\rm SPEX}(n,\mathcal{H})$ is a proper subset of ${\rm
EX}(n,\mathcal{H})$ in Theorem 1.1. To precisely characterize the graphs in
${\rm SPEX}(n,\mathcal{H})$, we need to develop new techniques.

Let $G$ be a graph. For an integer $\ell\geq 1$, a walk of length $\ell$ in
$G$ is an ordered sequence of vertices $v_{0},v_{1},...,v_{\ell}$, such that
$v_{i}\sim v_{i+1}$ for any $0\leq i\leq\ell-1$. The walk is called an
$\ell$-walk, and the vertex $v_{0}$ is called the starting vertex of the walk.
For a vertex $u$ of $G$, let $w^{\ell}_{G}(u)$ be the number of $\ell$-walks
starting at $u$ in $G$. Let $W^{\ell}(G)=\sum_{v\in V(G)}w^{\ell}_{G}(u)$.
Clearly, $w^{1}_{G}(u)=d_{G}(u)$ and

$w^{\ell+1}_{G}(u)=\sum_{v\in V(G),v\sim u}w^{\ell}_{G}(v)$

for any $\ell\geq 1$. Thus $W^{1}(G)=2|E(G)|$, and

$W^{2}(G)=\sum_{u\in V(G)}\sum_{v\in V(G),v\sim u}d_{G}(v)=\sum_{v\in
V(G)}d^{2}_{G}(v).$

Let $G_{1}$ and $G_{2}$ be two graphs (probably of different orders). We say
that $G_{1}$ is strictly walk-preferable to $G_{2}$ (denoted by $G_{1}\succ
G_{2}$), if there is an integer $\ell\geq 1$ such that
$W^{\ell}(G_{1})>W^{\ell}(G_{2})$ and $W^{i}(G_{1})=W^{i}(G_{2})$ for any
$1\leq i\leq\ell-1$; $G_{1}$ is walk-equivalent to $G_{2}$ (denoted by
$G_{1}\equiv G_{2}$), if $W^{i}(G_{1})=W^{i}(G_{2})$ for any $i\geq 1$. We say
$G_{1}$ is walk-preferable to $G_{2}$ (denoted by $G_{1}\succeq G_{2}$), if
$G_{1}\succ G_{2}$ or $G_{1}\equiv G_{2}$. We also write $G_{2}\prec G_{1}$ or
$G_{2}\preceq G_{1}$, if $G_{1}\succ G_{2}$ or $G_{1}\succeq G_{2}$. Let
$\mathcal{G}$ be a set of graphs. Let ${\rm SPEX}(\mathcal{G})$ denote the set
of graphs with the maximum spectral radius in $\mathcal{G}$. Define ${\rm
EX}^{1}(\mathcal{G})=\left\\{G\in\mathcal{G}~{}|~{}W^{1}(G)\geq
W^{1}(G^{\prime})~{}{\rm for~{}any}~{}G^{\prime}\in\mathcal{G}\right\\}$, and
for any $\ell\geq 2$ define ${\rm EX}^{\ell}(\mathcal{G})=\left\\{G\in{\rm
EX}^{\ell-1}(\mathcal{G})~{}|~{}W^{\ell}(G)\geq W^{\ell}(G^{\prime})~{}{\rm
for~{}any}~{}G^{\prime}\in{\rm EX}^{\ell-1}(\mathcal{G})\right\\}$. Clearly,
${\rm EX}^{1}(\mathcal{G})$ is precisely the set of graphs in $\mathcal{G}$
with the maximum number of edges, and ${\rm
EX}^{i+1}(\mathcal{G})\subseteq{\rm EX}^{i}(\mathcal{G})$ for any $i\geq 1$.
Let ${\rm EX}^{\infty}(\mathcal{G})=\cap_{1\leq i\leq\infty}{\rm
EX}^{i}(\mathcal{G})$. The graphs in ${\rm EX}^{\infty}(\mathcal{G})$ are
called the most walk-preferable graphs in $\mathcal{G}$.

The following Theorem 1.2 and Theorem 1.3 reveal the relationship between the
spectral radius of graphs and walks of their subgraphs.

###### Theorem 1.2

Let $G$ be a connected graph of order $n$, and let $S$ be a subset of $V(G)$
with $1\leq|S|<n$. Assume that $T$ is a set of some isolated vertices of
$G-S$, such that each vertex in $T$ is adjacent to each vertex in $S$ in $G$.
Let $H_{1}$ and $H_{2}$ be two graphs with vertex set $T$. For any $1\leq
i\leq 2$, let $G_{i}$ be the graph obtained from $G$ by embedding the edges of
$H_{i}$ into $T$. When $\rho(G)$ is sufficiently large (compared with $|T|$),
we have the following conclusions.
$(i)$ If $H_{1}\equiv H_{2}$, then $\rho(G_{1})=\rho(G_{2})$.
$(ii)$ If $H_{1}\succ H_{2}$, then $\rho(G_{1})>\rho(G_{2})$.
$(iii)$ If $H_{1}\prec H_{2}$, then $\rho(G_{1})<\rho(G_{2})$.

###### Theorem 1.3

For an integer $r\geq 2$, let $K_{n_{1},n_{2},...,n_{r}}$ be the complete
$r$-partite graph of order $n$ with parts $V_{1},V_{2},...,V_{r}$, where
$n_{i}=|V_{i}|\geq 1$ for $1\leq i\leq r$ and $\sum_{1\leq i\leq r}n_{i}=n$.
For each $1\leq i\leq r$, let $H_{i}$ be a graph with vertex set
$V(H_{i})\subseteq V_{i}$ and maximum degree at most $\Delta$. Let $G$ be the
graph obtained from $K_{n_{1},n_{2},...,n_{r}}$ by embedding the edges of
$H_{i}$ into $V_{i}$ for all $1\leq i\leq r$. Let $\rho=\rho(G)$. If
$\rho>\Delta$, then

$\sum_{1\leq s\leq
r}\frac{1}{1+\frac{n_{s}}{\rho}+\sum^{\infty}_{i=1}\frac{W^{i}(H_{s})}{\rho^{i+1}}}=r-1.$

To characterize the graphs in ${\rm SPEX}(n,\mathcal{H})$ in Theorem 1.1, we
first introduce some notations. For $r\geq 2$ and sufficiently large $n$, let
$T(n,r)$ be the Turán graph of order $n$ with $r$-parts
$V_{1},V_{2},...,V_{r}$, where $n_{i}=|V_{i}|$ for $1\leq i\leq r$ and
$\lfloor\frac{n}{r}\rfloor=n_{1}\leq n_{2}\leq\cdots\leq
n_{r}=\lceil\frac{n}{r}\rceil$. For a fixed integer $t\geq 1$, let
$\mathcal{T}_{n,r,t}$ be the set of all the graphs obtained from $T(n,r)$ by
embedding $t$ edges into the $r$-parts $V_{1},V_{2},...,V_{r}$. Given a graph
$G\in\mathcal{T}_{n,r,t}$, let $H^{G}_{i}$ (probably empty) be the subgraph of
$G$ induced by the edges embedded into $V_{i}$ for each $1\leq i\leq r$.
Clearly, $H^{G}_{1},H^{G}_{2},...,H^{G}_{r}$ are vertex-disjoint. Let
$H^{G}=\cup_{1\leq i\leq r}H^{G}_{i}$. Then $|E(H^{G})|=\sum_{1\leq i\leq
r}|E(H^{G}_{i})|=t$.

For a subset $\mathcal{G}$ of $\mathcal{T}_{n,r,t}$, let
$\mathcal{G}_{j}=\left\\{G\in\mathcal{G}~{}|~{}H^{G}=\cup_{1\leq i\leq
r}H^{G}_{i}=H^{G}_{j}\right\\}$ for any $1\leq j\leq r$. Define
$\mathcal{H}^{\mathcal{G}}=\left\\{H^{G}~{}|~{}G\in\mathcal{G}\right\\}$ and
$\mathcal{H}^{\mathcal{G}_{j}}=\left\\{H^{G}~{}|~{}G\in\mathcal{G}_{j}\right\\}$
(if $\mathcal{G}_{j}$ is not empty) for $1\leq j\leq r$. The subset
$\mathcal{G}$ is called a normal set of $\mathcal{T}_{n,r,t}$, if it satisfies
the property: for any $1\leq i\neq j\leq r$ and any $G\in\mathcal{G}_{i}$,
there is a $G^{\prime}\in\mathcal{G}_{j}$ such that
$H^{G^{\prime}}=H^{G^{\prime}}_{j}$ is a copy of $H^{G}=H^{G}_{i}$.

###### Theorem 1.4

For $t\geq 1,r\geq 2$ and sufficiently large $n$, let $\mathcal{G}$ be a
normal subset of $\mathcal{T}_{n,r,t}$. Let
$\mathcal{G}_{i},\mathcal{H^{\mathcal{G}}}$ and
$\mathcal{H}^{\mathcal{G}_{i}}$ be defined as above for any $1\leq i\leq r$.
If $\left\\{G\in\mathcal{G}~{}|~{}H^{G}\in{\rm
EX}^{2}(\mathcal{H}^{\mathcal{G}})\right\\}\subseteq\cup_{1\leq i\leq
r}\mathcal{G}_{i}$, then
$(i)$ ${\rm SPEX}(\mathcal{G})=\cup_{i,n_{i}=\lfloor\frac{n}{r}\rfloor}{\rm
SPEX}(\mathcal{G}_{i})$;
$(ii)$ for any $1\leq i\leq r$ and any graph $G\in\mathcal{G}_{i}$, $G\in{\rm
SPEX}(\mathcal{G}_{i})$ if and only if $H^{G}\in{\rm
EX}^{\infty}(\mathcal{H}^{\mathcal{G}_{i}})$.

Considering Theorem 1.2 and Theorem 1.4, it is interesting to study the
following problem:

Problem. For a specified family $\mathcal{H}$ of graphs, characterize the
graphs in ${\rm EX}^{\infty}(\mathcal{H})$.

Recall that $W^{2}(G)=\sum_{v\in V(G)}d^{2}_{G}(u)$ for any graph $G$. The
following result is taken from Lemma 2.4 of [10] (it is trivial for $m=1$).

###### Lemma 1.5

(Ismailescu and Stefanica [10]) Let $\mathcal{M}_{n,m}$ be the set of all the
graphs of order $n$ with $m$ edges, where $m\geq 1$ and $n\geq m+2$. Then
${\rm
EX}^{2}(\mathcal{M}_{n,m})=\left\\{K_{1,3}\cup\overline{K_{n-4}},K_{3}\cup\overline{K_{n-3}}\right\\}$
for $m=3$, and ${\rm
EX}^{2}(\mathcal{M}_{n,m})=\left\\{K_{1,m}\cup\overline{K_{n-1-m}}\right\\}$
otherwise.

It is easy to see that $W^{3}(K_{3})=24>18=W^{3}(K_{1,3})$. Since a graph of
order less than $n$ can be viewed as a graph of order $n$ by adding some
isolated vertices, the following result can be deduced from Lemma 1.5 directly
(by letting $n=2m+2$).

###### Corollary 1.6

Let $\mathcal{M}_{m}$ be the set of graphs with $m$ edges and without isolated
vertices, where $m\geq 1$. Then we have the following conclusions.
$(i)$ ${\rm EX}^{2}(\mathcal{M}_{m})=\left\\{K_{1,3},K_{3}\right\\}$ for
$m=3$, and ${\rm EX}^{2}(\mathcal{M}_{m})=\left\\{K_{1,m}\right\\}$ otherwise
$(ii)$ ${\rm EX}^{\infty}(\mathcal{M}_{m})={\rm
EX}^{3}(\mathcal{M}_{m})=\left\\{K_{3}\right\\}$ for $m=3$, and ${\rm
EX}^{\infty}(\mathcal{M}_{m})={\rm
EX}^{3}(\mathcal{M}_{m})=\left\\{K_{1,m}\right\\}$ otherwise.

For a graph $G$, the maximum average degree of $G$ is the maximum value of
$\frac{2|E(H)|}{|H|}$, where $H$ is an induced subgraph of $G$ with $|H|\geq
1$. For $k\geq 1$ and $n\geq 2k+1$, Let $\mathcal{N}_{n,k}$ be the set of
graphs of order $n$ with maximum average degree at most $2k$. Zhang [20]
proved that any graph $G\in{\rm SPEX}(\mathcal{N}_{n,k})$ is obtained from
$K_{k}\vee\overline{K_{n-k}}$ by embedding $\frac{k(k+1)}{2}$ edges into the
part $V(\overline{K_{n-k}})$. When $k$ is given and $n$ is sufficiently large,
from Theorem 1.2 together with Corollary 1.6, we see that such $G$ is unique,
of which the subgraph induced by the $\frac{k(k+1)}{2}$ edges is $K_{3}$ for
$k=2$, and is $K_{1,\frac{k(k+1)}{2}}$ otherwise. (Note that, we can assume
that the $\frac{k(k+1)}{2}$ edges are embedded into a fixed subset of size
$k(k+1)$ which is viewed as $T$ in Theorem 1.2.)

A graph $F$ is called color-critical if there is an edge $e$ whose deletion
induces a subgraph with less chromatic number than $F$. As in [9], for $k\geq
2$ color-critical graphs with chromatic number $r+1\geq 3$:
$F_{1},F_{2},...,F_{k}$, let $\mathcal{G}(F_{1},F_{2},...,F_{k})$ denote the
family of graphs which consists of $k$ edge-disjoint copies of
$F_{1},F_{2},...,F_{k}$. Fang, Tait and Zhai [9] proved that for sufficiently
large $n$, ${\rm EX}(n,\mathcal{G}(F_{1},F_{2},...,F_{k}))$ is precisely
$\mathcal{T}_{n,r,k-1}$ defined as above. From Theorem 1.1 we see ${\rm
SPEX}(n,\mathcal{G}(F_{1},F_{2},...,F_{k}))\subseteq{\rm
EX}(n,\mathcal{G}(F_{1},F_{2},...,F_{k}))=\mathcal{T}_{n,r,k-1}$. The authors
in [9] also devoted themselves to characterize the graphs with the maximum
spectral radius in $\mathcal{T}_{n,r,k-1}$ (this result is given by Lin, Zhai
and Zhai [14] for the case $F_{1}=F_{2}=\cdots=F_{k}=K_{3}$). In detail, they
[9] proved the following Corollary 1.7, which is directly deduced from Theorem
1.4 together with Corollary 1.6. (The condition in Theorem 1.4 holds for
$\mathcal{G}=\mathcal{T}_{n,r,k-1}$. Clearly, $\mathcal{T}_{n,r,k-1}$ itself
is a normal subset of $\mathcal{T}_{n,r,k-1}$. Moreover, the condition

$\left\\{G\in\mathcal{T}_{n,r,k-1}~{}|~{}H^{G}\in{\rm
EX}^{2}(\mathcal{H}^{\mathcal{T}_{n,r,k-1}})\right\\}\subseteq\cup_{1\leq
i\leq r}\mathcal{G}_{i}$

holds, since the one or two graphs in ${\rm
EX}^{2}(\mathcal{H}^{\mathcal{T}_{n,r,k-1}})$ are connected by Corollary 1.6.)

###### Corollary 1.7

([9, 14]) For $k\geq 2,r\geq 2$ and sufficiently large $n$, let
$\mathcal{T}_{n,r,k-1}$ be defined as above. Then each graph in ${\rm
SPEX}(\mathcal{T}_{n,r,k-1})$ is obtained from $T_{n,r}$ by embedding $k-1$
edges into its one part of size $\lfloor\frac{n}{r}\rfloor$, of which the
subgraph induced by the $k-1$ edges is $K_{3}$ for $k=4$ and is $K_{1,k-1}$
otherwise.

##  2 Proofs of Theorem 1.2 and Theorem 1.3

The following Lemma 2.1 is taken from Theorem 8.1.1 of [2].

###### Lemma 2.1

([2]) If $H$ is a subgraph of a connected graph $G$, then
$\rho(H)\leq\rho(G)$, with equality if and only if $H=G$.

###### Lemma 2.2

Let $G$ be a connected graph of order $n$, and let $S$ be a subset of $V(G)$
with $1\leq|S|<n$. Assume that $H$ is a component of $G-S$ with maximum degree
at most $\Delta$, such that each vertex of $H$ is adjacent to each vertex in
$S$. Let $\rho=\rho(G)$ and let $\mathbf{x}=(x_{v})_{v\in V(G)}$ be the Perron
vector of $G$ such that $\sum_{v\in S}x_{v}=\rho$. If $\rho>\Delta$, then for
any $u\in T$ we have

$x_{u}=1+\sum^{\infty}_{i=1}\frac{w_{H}^{i}(u)}{\rho^{i}}.$

Proof: Denote $V(H)=T$. Recall that $\sum_{v\in S}x_{v}=\rho$. For any $u\in
T$, we have

$\rho x_{u}=(\sum_{v\in S}x_{v})+\sum_{v\sim u,v\in T}x_{v}=\rho+\sum_{v\sim
u,v\in T}x_{v}.$

This implies that $x_{u}=1+\frac{\sum_{v\sim u,v\in T}x_{v}}{\rho}$.

Claim 1. For any vertex $u$ in $T$, we have $x_{u}\geq 1+\sum_{1\leq
i\leq\ell}\frac{w_{H}^{i}(u)}{\rho^{i}}$ for any integer $\ell\geq 1$.

Proof of Claim 1. From $x_{u}=1+\frac{\sum_{v\sim u,v\in T}x_{v}}{\rho}$ we
see that $x_{u}\geq 1$ for any $u\in T$. Thus, for any $v\in T$ and $v\sim u$
we have $x_{v}\geq 1$. By $x_{u}=1+\frac{\sum_{v\sim u,v\in T}x_{v}}{\rho}$ we
have $x_{u}\geq 1+\frac{w_{H}^{1}(u)}{\rho}$ for any $u\in T$. Thus Claim 1
holds for $\ell=1$. Assume that Claim 1 holds for $\ell=j\geq 1$. That is
$x_{u}\geq 1+\sum_{1\leq i\leq j}\frac{w_{H}^{i}(u)}{\rho^{i}}$ for any $v\in
T$. Now we consider the case $\ell=j+1$. Let $u$ be a vertex in $T$. Note that
$x_{v}\geq 1+\sum_{1\leq i\leq j}\frac{w_{H}^{i}(v)}{\rho^{i}}$ for any $v\in
T$ and $v\sim u$ by induction on $\ell=j$. By $x_{u}=1+\frac{\sum_{v\sim
u,v\in T}x_{v}}{\rho}$, we have

$x_{u}\geq 1+\frac{\sum_{v\sim u,v\in T}(1+\sum_{1\leq i\leq
j}\frac{w_{H}^{i}(v)}{\rho^{i}})}{\rho}=1+\sum_{1\leq i\leq
j+1}\frac{w_{H}^{i}(u)}{\rho^{i}}.$

So, Claim 1 holds for $\ell=j+1$. By induction on $\ell$, we complete the
proof of Claim 1. $\Box$

Claim 2. For any vertex $u$ in $T$, we have $x_{u}\leq(1+\sum_{1\leq
i\leq\ell}\frac{w_{H}^{i}(u)}{\rho^{i}})+\frac{\Delta}{\rho-\Delta}\frac{w_{H}^{\ell}(u)}{\rho^{\ell}}$
for any integer $\ell\geq 1$.

Proof of Claim 2. Let $u^{*}$ be a vertex in $T$ such that
$x_{u^{*}}=\max_{v\in T}x_{v}$. Recall that $\rho>\Delta$. From
$x_{u^{*}}=1+\frac{\sum_{v\sim u^{*},v\in T}x_{v}}{\rho}\leq
1+\frac{\Delta\cdot x_{u^{*}}}{\rho}$ we see that
$x_{u^{*}}\leq\frac{\rho}{\rho-\Delta}$.

For any vertex $u\in T$, from $x_{u}=1+\frac{\sum_{v\sim u,v\in
T}x_{v}}{\rho}$ we obtain that

$x_{u}\leq 1+\frac{w_{H}^{1}(u)\cdot
x_{u^{*}}}{\rho}\leq(1+\frac{w_{H}^{1}(u)}{\rho})+\frac{\Delta}{\rho-\Delta}\frac{w_{H}^{1}(u)}{\rho}.$

Thus Claim 2 holds for $\ell=1$. Assume that Claim 2 holds for $\ell=j\geq 1$.
That is $x_{u}\leq(1+\sum_{1\leq i\leq
j}\frac{w_{H}^{i}(u)}{\rho^{i}})+\frac{\Delta}{\rho-\Delta}\frac{w_{H}^{j}(u)}{\rho^{j}}$
for any $u\in T$. Now we consider the case $\ell=j+1$. Let $u\in T$. Note that
$x_{v}\leq(1+\sum_{1\leq i\leq
j}\frac{w_{H}^{i}(v)}{\rho^{i}})+\frac{\Delta}{\rho-\Delta}\frac{w_{H}^{j}(v)}{\rho^{j}}$
for any $v\in T$ and $v\sim u$ by induction on $\ell=j$. By
$x_{u}=1+\frac{\sum_{v\sim u,v\in T}x_{v}}{\rho}$, we have

$x_{u}\leq 1+\frac{\sum_{v\sim u,v\in T}((1+\sum_{1\leq i\leq
j}\frac{w_{H}^{i}(v)}{\rho^{i}})+\frac{\Delta}{\rho-\Delta}\frac{w_{H}^{j}(v)}{\rho^{j}})}{\rho}=(1+\sum_{1\leq
i\leq
j+1}\frac{w_{H}^{i}(u)}{\rho^{i}})+\frac{\Delta}{\rho-\Delta}\frac{w_{H}^{j+1}(u)}{\rho^{j+1}}.$

So, Claim 2 holds for $\ell=j+1$. By induction on $\ell$, we complete the
proof of Claim 2. $\Box$

Since $\rho>\Delta$, we see that the infinite series
$\sum^{\infty}_{i=1}(\frac{\Delta}{\rho})^{i}$ exists and
$(\frac{\Delta}{\rho})^{\ell}\rightarrow 0$ for $\ell\rightarrow\infty$. Let
$u\in T$. For any $\ell\geq 1$, clearly, $w_{H}^{\ell}(u)\leq\Delta^{\ell}$.
Thus $\sum^{\infty}_{i=1}\frac{w_{H}^{i}(u)}{\rho^{i}}$ exists and
$\frac{w_{H}^{\ell}(u)}{\rho^{\ell}}\rightarrow 0$ for
$\ell\rightarrow\infty$. By Claim 1 and Claim 2 we have

$1+\sum_{1\leq i\leq\ell}\frac{w_{H}^{i}(u)}{\rho^{i}}\leq
x_{u}\leq(1+\sum_{1\leq
i\leq\ell}\frac{w_{H}^{i}(u)}{\rho^{i}})+\frac{\Delta}{\rho-\Delta}\frac{w_{H}^{\ell}(u)}{\rho^{\ell}},$

for any integer $\ell\geq 1$. By letting $\ell\rightarrow\infty$ we obtain
that $x_{u}=1+\sum^{\infty}_{i=1}\frac{w_{H}^{i}(u)}{\rho^{i}}$. This
completes the proof. $\Box$

Proof of Theorem 1.2. Clearly, both $G_{1}$ and $G_{2}$ are connected. Let
$\rho_{1}=\rho(G_{1})$ and $\rho_{2}=\rho(G_{2})$. Then
$\rho_{1},\rho_{2}\geq\rho(G)$ by Lemma 2.1. Let $\mathbf{x}=(x_{v})_{v\in
V(G_{1})}$ be a Perron vector of $G_{1}$ such that $\sum_{v\in
S}x_{v}=\rho_{1}$. Let $\mathbf{y}=(y_{v})_{v\in V(G_{2})}$ be a Perron vector
of $G_{2}$ such that $\sum_{v\in S}x_{v}=\rho_{2}$.

Claim 1. $(\rho_{1}-\rho_{2})\sum_{v\in
V(G)}x_{v}y_{v}=\sum^{\infty}_{\ell=0}(\frac{W^{\ell+1}(H_{1})}{\rho_{1}^{\ell}}-\frac{W^{\ell+1}(H_{2})}{\rho_{2}^{\ell}})$.

Proof of Claim 1. Clearly, the only distinct edges of $G_{1}$ and $G_{2}$ are
inside $T$. Let $A(G_{1})$ and $A(G_{2})$ be the adjacency matrices of $G_{1}$
and $G_{2}$, respectively. Let $\mathbf{x}^{{}^{\prime}}$ be the transpose of
$\mathbf{x}$. For any $u\in T$, we write $w^{0}_{H_{i}}(u)=1$ for any $1\leq
i\leq 2$. Thus
$x_{u}=\sum^{\infty}_{i=0}\frac{w_{H_{1}}^{i}(u)}{\rho_{1}^{i}}$ and
$y_{u}=\sum^{\infty}_{i=0}\frac{w_{H_{2}}^{i}(u)}{\rho_{2}^{i}}$ by Lemma 2.2.
Then

$\displaystyle(\rho_{1}-\rho_{2})\sum_{v\in V(G)}x_{v}y_{v}$
$\displaystyle=\mathbf{x}^{{}^{\prime}}(A(G_{1})-A(G_{2}))\mathbf{y}$
$\displaystyle=(\sum_{uv\in E(H_{1})}(x_{u}y_{v}+x_{v}y_{u}))-\sum_{uv\in
E(H_{2})}(x_{u}y_{v}+x_{v}y_{u})$ $\displaystyle=(\sum_{u\in T}y_{u}\sum_{v\in
T,vu\in E(H_{1})}x_{v})-\sum_{u\in T}x_{u}\sum_{v\in T,vu\in E(H_{2})}y_{v}$
$\displaystyle=(\sum_{u\in
T}(\sum^{\infty}_{i=0}\frac{w_{H_{2}}^{i}(u)}{\rho_{2}^{i}})(\sum_{v\in
T,vu\in
E(H_{1})}\sum^{\infty}_{i=0}\frac{w_{H_{1}}^{i}(v)}{\rho_{1}^{i}}))-\sum_{u\in
T}(\sum^{\infty}_{i=0}\frac{w_{H_{1}}^{i}(u)}{\rho_{1}^{i}})(\sum_{v\in
T,vu\in E(H_{2})}\sum^{\infty}_{i=0}\frac{w_{H_{2}}^{i}(v)}{\rho_{2}^{i}})$
$\displaystyle=(\sum_{u\in
T}(\sum^{\infty}_{i=0}\frac{w_{H_{2}}^{i}(u)}{\rho_{2}^{i}})(\sum^{\infty}_{i=0}\frac{w_{H_{1}}^{i+1}(u)}{\rho_{1}^{i}}))-\sum_{u\in
T}(\sum^{\infty}_{i=0}\frac{w_{H_{1}}^{i}(u)}{\rho_{1}^{i}})(\sum^{\infty}_{i=0}\frac{w_{H_{2}}^{i+1}(u)}{\rho_{2}^{i}})$
$\displaystyle=(\sum_{u\in T}\sum^{\infty}_{\ell=0}\sum_{i,j\geq
0,i+j=\ell}\frac{w_{H_{1}}^{i+1}(u)w_{H_{2}}^{j}(u)}{\rho_{1}^{i}\rho_{2}^{j}})-\sum_{u\in
T}\sum^{\infty}_{\ell=0}\sum_{i,j\geq
0,i+j=\ell}\frac{w_{H_{1}}^{i}(u)w_{H_{2}}^{j+1}(u)}{\rho_{1}^{i}\rho_{2}^{j}}$
$\displaystyle=\sum_{u\in
T}\sum^{\infty}_{\ell=0}(\frac{w_{H_{1}}^{\ell+1}(u)w_{H_{2}}^{0}(u)}{\rho_{1}^{\ell}\rho_{2}^{0}}-\frac{w_{H_{1}}^{0}(u)w_{H_{2}}^{\ell+1}(u)}{\rho_{1}^{0}\rho_{2}^{\ell}})$
$\displaystyle=\sum^{\infty}_{\ell=0}(\frac{W^{\ell+1}(H_{1})}{\rho_{1}^{\ell}}-\frac{W^{\ell+1}(H_{2})}{\rho_{2}^{\ell}}).$

This finishes the proof of Claim 1. $\Box$

Clearly, there are at most $2^{\frac{1}{2}|T|^{2}}$ graphs with vertex set
$T$. Thus there are at most
$\frac{1}{2}(2^{\frac{1}{2}|T|^{2}})^{2}=2^{|T|^{2}-1}$ pairs of graphs
$Q_{1},Q_{2}$ with vertex set $T$ such that $Q_{1}\succ Q_{2}$. Let
$\ell_{Q_{1},Q_{2}}$ be the integer such that
$W^{\ell_{Q_{1},Q_{2}}}(Q_{1})>W^{\ell_{Q_{1},Q_{2}}}(Q_{2})$ and
$W^{i}(Q_{1})=W^{i}(Q_{2})$ for any $1\leq i\leq\ell_{Q_{1},Q_{2}}-1$. Let
$C=\max_{Q_{1}\succ Q_{2},V(Q_{1})=V(Q_{2})=T}\ell_{Q_{1},Q_{2}}$. Note that
$C$ is only related to $|T|$. When $\rho(G)$ is sufficiently large, we can
assume $\rho(G)>|T|-1+(|T|-1)^{C+1}|T|$ in the following discussion.

$(i)$ Let $H_{1}\equiv H_{2}$. Then $W^{i}(H_{1})=W^{i}(H_{2})$ for any $i\geq
1$. Without loss of generality, assume $\rho_{1}\geq\rho_{2}$. Then by Claim 1
we have

$(\rho_{1}-\rho_{2})\sum_{v\in
V(G)}x_{v}y_{v}=\sum^{\infty}_{\ell=0}(\frac{W^{\ell+1}(H_{1})}{\rho_{1}^{\ell}}-\frac{W^{\ell+1}(H_{2})}{\rho_{2}^{\ell}})\leq
0.$

It follows that $\rho_{1}\leq\rho_{2}$. Consequently, $\rho_{1}=\rho_{2}$.

$(ii)$ Let $H_{1}\succ H_{2}$. There is an integer $1\leq k\leq C$ (defined as
above), such that $W^{k}(H_{1})>W^{k}(H_{2})$ and $W^{i}(H_{1})=W^{i}(H_{2})$
for any $1\leq i\leq k-1$. We shall prove $\rho_{1}>\rho_{2}$. Suppose
$\rho_{1}\leq\rho_{2}$ by contradiction. Then by Claim 1 we have

$\displaystyle(\rho_{1}-\rho_{2})\sum_{v\in V(G)}x_{v}y_{v}$
$\displaystyle=\sum^{\infty}_{\ell=0}(\frac{W^{\ell+1}(H_{1})}{\rho_{1}^{\ell}}-\frac{W^{\ell+1}(H_{2})}{\rho_{2}^{\ell}})$
$\displaystyle\geq\sum^{\infty}_{\ell=0}(\frac{W^{\ell+1}(H_{1})}{\rho_{2}^{\ell}}-\frac{W^{\ell+1}(H_{2})}{\rho_{2}^{\ell}})$
$\displaystyle=\sum^{\infty}_{\ell=k-1}(\frac{W^{\ell+1}(H_{1})}{\rho_{2}^{\ell}}-\frac{W^{\ell+1}(H_{2})}{\rho_{2}^{\ell}})$
$\displaystyle\geq\frac{1}{\rho^{k-1}_{2}}-\sum^{\infty}_{\ell=k}\frac{W^{\ell+1}(H_{2})}{\rho_{2}^{\ell}}$
$\displaystyle\geq\frac{1}{\rho^{k-1}_{2}}-\sum^{\infty}_{\ell=k}\frac{(|T|-1)^{\ell+1}|T|}{\rho_{2}^{\ell}}$
$\displaystyle=\frac{1}{\rho^{k-1}_{2}(\rho_{2}-|T|+1)}(\rho_{2}-|T|+1-(|T|-1)^{k+1}|T|)$
$\displaystyle\geq\frac{1}{\rho^{k-1}_{2}(\rho_{2}-|T|+1)}(\rho_{2}-|T|+1-(|T|-1)^{C+1}|T|)>0.$

It follows that $\rho_{1}>\rho_{2}$, which contradicts the assumption
$\rho_{1}\leq\rho_{2}$. Hence we have $\rho_{1}>\rho_{2}$.

$(iii)$ The proof is very similar to $(ii)$. This completes the proof. $\Box$

Proof of Theorem 1.3. Let $\mathbf{x}=(x_{v})_{v\in V(G)}$ be the Perron
vector of $G$ such that $\sum_{v\in V(G)}x_{v}=1$. For each $1\leq s\leq r$,
let $\mathbf{y}^{s}=\frac{\rho}{1-\sum_{v\in V_{s}}x_{v}}\mathbf{x}$. Then
$\mathbf{y}^{s}=(y^{s}_{v})_{v\in V(G)}$ is a Perron vector of $G$ such that

$\sum_{v\in V(G)-V_{s}}y^{s}_{v}=\frac{\rho}{1-\sum_{v\in
V_{s}}x_{v}}\sum_{v\in V(G)-V_{s}}x_{v}=\rho.$

By Lemma 2.2, we have

$y^{s}_{u}=1+\sum^{\infty}_{i=1}\frac{w_{H_{s}}^{i}(u)}{\rho^{i}},$

for any $u\in V_{s}$. It follows that

$x_{u}=\frac{1-\sum_{v\in
V_{s}}x_{v}}{\rho}(1+\sum^{\infty}_{i=1}\frac{w_{H_{s}}^{i}(u)}{\rho^{i}}),$

for any $u\in V_{s}$. Thus

$\sum_{u\in V_{s}}x_{u}=\frac{1-\sum_{v\in V_{s}}x_{v}}{\rho}\sum_{u\in
V_{s}}(1+\sum^{\infty}_{i=1}\frac{w_{H_{s}}^{i}(u)}{\rho^{i}})=\frac{1-\sum_{u\in
V_{s}}x_{u}}{\rho}(n_{s}+\sum^{\infty}_{i=1}\frac{W^{i}(H_{s})}{\rho^{i}}).$

It follows that

$\sum_{u\in
V_{s}}x_{u}=1-\frac{1}{1+\frac{n_{s}}{\rho}+\sum^{\infty}_{i=1}\frac{W^{i}(H_{s})}{\rho^{i+1}}}.$

Then

$1=\sum_{1\leq s\leq r}\sum_{u\in V_{s}}x_{u}=r-\sum_{1\leq s\leq
r}\frac{1}{1+\frac{n_{s}}{\rho}+\sum^{\infty}_{i=1}\frac{W^{i}(H_{s})}{\rho^{i+1}}},$

implying that

$\sum_{1\leq s\leq
r}\frac{1}{1+\frac{n_{s}}{\rho}+\sum^{\infty}_{i=1}\frac{W^{i}(H_{s})}{\rho^{i+1}}}=r-1.$

This completes the proof. $\Box$

##  3 Proof of Theorem 1.4

In this section, we prove Theorem 1.4 by using Theorem 1.2 and Theorem 1.3.
For $t\geq 1,r\geq 2$ and sufficiently large $n$, let $\mathcal{G}$ be a
normal subset of $\mathcal{T}_{n,r,t}$. Let
$\mathcal{G}_{i},\mathcal{H^{\mathcal{G}}}$ and
$\mathcal{H}^{\mathcal{G}_{i}}$ for $1\leq i\leq r$ be defined as in the
introduction.

Proof of Theorem 1.4. For any $G\in\mathcal{G}$, we have
$\rho(G)\geq\rho(T(n,r))$ by Lemma 2.1, and
$\rho(G)\leq\rho(T(n,r))+\rho(H^{G})$ by Courant-Weyl inequalities (see
Theorem 1.3.15 of [2]). As we know,
$\rho(H^{G})\leq\sqrt{2|E(H^{G})|}=\sqrt{2t}$. Thus
$\rho(T(n,r))\leq\rho(G)\leq\rho(T(n,r))+\sqrt{2t}$. Note that
$\frac{\rho(T(n,r))}{n}\rightarrow 1-\frac{1}{r}$ for $n\rightarrow\infty$.
Thus for any $1\leq s\leq r$, we have
$\frac{n_{s}}{\rho(G)}\rightarrow\frac{\frac{1}{r}}{1-\frac{1}{r}}=\frac{1}{r-1}$
for $n\rightarrow\infty$. Let $\rho=\rho(G)$. Then by Theorem 1.3 we have

$\sum_{1\leq s\leq
r}\frac{1}{1+\frac{n_{s}}{\rho}+\sum^{\infty}_{i=1}\frac{W^{i}(H^{G}_{s})}{\rho^{i+1}}}=r-1.$

Let

$f(G,x)=\sum_{1\leq s\leq
r}\frac{1}{1+\frac{n_{s}}{x}+\sum^{\infty}_{i=1}\frac{W^{i}(H^{G}_{s})}{x^{i+1}}},$

where $x>0$ is large enough such that
$\sum^{\infty}_{i=1}\frac{W^{i}(H^{G}_{s})}{x^{i+1}}$ exists for $1\leq s\leq
r$. Clearly, $f(G,x)$ is strictly increasing with respective to $x$. Thus
$\rho$ is the largest root of $f(G,x)=r-1$. Note that
$\frac{1}{1+x}=1+\sum_{i\geq 1}(-x)^{i}$ for $0\leq x<1$. Thus for $1\leq
s\leq r$, (noting that $W^{i}(H^{G}_{s})\leq 2t\cdot t^{i}$ for any $i\geq 1$)
we have

$\displaystyle\frac{1}{1+\frac{n_{s}}{\rho}+\sum^{\infty}_{i=1}\frac{W^{i}(H^{G}_{s})}{\rho^{i+1}}}$
$\displaystyle=\frac{1}{1+\frac{n_{s}}{\rho}}\frac{1}{1+\frac{1}{1+\frac{n_{s}}{\rho}}\sum^{\infty}_{i=1}\frac{W^{i}(H^{G}_{s})}{\rho^{i+1}}}$
$\displaystyle=\frac{1}{1+\frac{n_{s}}{\rho}}(1-\frac{1}{1+\frac{n_{s}}{\rho}}\frac{W^{1}(H^{G}_{s})}{\rho^{2}}-\frac{1}{1+\frac{n_{s}}{\rho}}\frac{W^{2}(H^{G}_{s})}{\rho^{3}}+\Theta(\frac{1}{\rho^{4}})).$

Then

$f(G,x)=(\sum_{1\leq s\leq
r}(\frac{1}{1+\frac{n_{s}}{x}}-\frac{1}{(1+\frac{n_{s}}{x})^{2}}\frac{W^{1}(H^{G}_{s})}{x^{2}}-\frac{1}{(1+\frac{n_{s}}{x})^{2}}\frac{W^{2}(H^{G}_{s})}{x^{3}}))+\Theta(\frac{1}{x^{4}}).$

Without loss of generality, assume $G\in\rm SPEX(\mathcal{G})$. Let
$G^{\prime}\in\mathcal{G}$ such that $H^{G^{\prime}}\in\rm
EX^{2}(\mathcal{H}^{\mathcal{G}})$. Then $G^{\prime}\in\cup_{1\leq i\leq
r}\mathcal{G}_{i}$ by the condition. Since $\mathcal{G}$ is a normal subset of
$\mathcal{T}_{n,r,t}$, we can assume that $G^{\prime}\in\mathcal{G}_{1}$. Note
that $\rho=\rho(G)\geq\rho(G^{\prime})$. Since $f(G^{\prime},x)$ is strictly
increasing with respective to $x$ and $f(G^{\prime},\rho(G^{\prime}))=r-1$, we
have $f(G^{\prime},\rho)\geq r-1$. It follows that $f(G^{\prime},\rho)\geq
f(G,\rho)$ as $f(G,\rho)=r-1$. Thus
$\rho^{3}(f(G^{\prime},\rho)-f(G,\rho))\geq 0$. On the other hand, we have

$\displaystyle\rho^{3}(f(G^{\prime},\rho)-f(G,\rho))$
$\displaystyle=\rho^{3}(\sum_{1\leq s\leq
r}(\frac{1}{1+\frac{n_{s}}{\rho}}-\frac{1}{(1+\frac{n_{s}}{\rho})^{2}}\frac{W^{1}(H^{G^{\prime}}_{s})}{\rho^{2}}-\frac{1}{(1+\frac{n_{s}}{\rho})^{2}}\frac{W^{2}(H^{G^{\prime}}_{s})}{\rho^{3}}))+\Theta(\frac{1}{\rho})$
$\displaystyle-\rho^{3}(\sum_{1\leq s\leq
r}(\frac{1}{1+\frac{n_{s}}{\rho}}-\frac{1}{(1+\frac{n_{s}}{\rho})^{2}}\frac{W^{1}(H^{G}_{s})}{\rho^{2}}-\frac{1}{(1+\frac{n_{s}}{\rho})^{2}}\frac{W^{2}(H^{G}_{s})}{\rho^{3}}))-\Theta(\frac{1}{\rho})$
$\displaystyle=(\rho\sum_{1\leq s\leq
r}\frac{W^{1}(H^{G}_{s})-W^{1}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}})+(\sum_{1\leq
s\leq
r}\frac{W^{2}(H^{G}_{s})-W^{2}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}})+\Theta(\frac{1}{\rho}).$

Since $G^{\prime}\in\mathcal{G}_{1}$, we have that
$H^{G^{\prime}}=H^{G^{\prime}}_{1}$, i.e., $H^{G^{\prime}}_{i}$ is empty for
$2\leq i\leq r$. Thus $W^{1}(H^{G^{\prime}}_{1})=\sum_{1\leq s\leq
r}W^{1}(H^{G}_{s})=2t$, implying that $\sum_{1\leq s\leq
r}\frac{W^{1}(H^{G}_{s})-W^{1}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}}\leq
0$ (recall $n_{1}=\lfloor\frac{n}{r}\rfloor$). From
$\rho^{3}(f(G^{\prime},\rho)-f(G,\rho))\geq 0$, we obtain that

$(\sum_{1\leq s\leq
r}\frac{W^{2}(H^{G}_{s})-W^{2}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}})+\Theta(\frac{1}{\rho})\geq
0.$

Note that $W^{2}(H^{G^{\prime}}_{1})\geq\sum_{1\leq s\leq
r}W^{2}(H^{G}_{s})=W^{2}(H^{G})$ as $H^{G^{\prime}}\in\rm
EX^{2}(\mathcal{H}^{\mathcal{G}})$. If $W^{2}(H^{G^{\prime}}_{1})>\sum_{1\leq
s\leq r}W^{2}(H^{G}_{s})$, then for $n\rightarrow\infty$,

$\sum_{1\leq s\leq
r}\frac{W^{2}(H^{G}_{s})-W^{2}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}}\leq-\frac{1}{(1+\frac{n_{1}}{\rho})^{2}}\rightarrow-\frac{1}{(1+\frac{1}{r-1})^{2}}.$

This contradicts that $(\sum_{1\leq s\leq
r}\frac{W^{2}(H^{G}_{s})-W^{2}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}})+\Theta(\frac{1}{\rho})\geq
0$. Hence

$W^{2}(H^{G^{\prime}}_{1})=\sum_{1\leq s\leq r}W^{2}(H^{G}_{s})=W^{2}(H^{G}).$

Recall that $H^{G^{\prime}}\in\rm EX^{2}(\mathcal{H}^{\mathcal{G}})$. Thus
$H^{G}\in\rm EX^{2}(\mathcal{H}^{\mathcal{G}})$, implying that
$G\in\cup_{1\leq i\leq r}\mathcal{G}_{i}$ by the condition of the theorem.

Suppose that $G\in\mathcal{G}_{j}$ for some $1\leq j\leq r$ and
$n_{j}=\lceil\frac{n}{r}\rceil>\lfloor\frac{n}{r}\rfloor$. Then for
$n\rightarrow\infty$,

$\rho\sum_{1\leq s\leq
r}\frac{W^{1}(H^{G}_{s})-W^{1}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}}=t\rho(\frac{1}{(1+\frac{n_{j}}{\rho})^{2}}-\frac{1}{(1+\frac{n_{1}}{\rho})^{2}})=\frac{-t(2+\frac{n_{j}}{\rho}+\frac{n_{1}}{\rho})}{(1+\frac{n_{j}}{\rho})^{2}(1+\frac{n_{1}}{\rho})^{2}}$
$\rightarrow\frac{-t(2+\frac{2}{r-1})}{(1-\frac{1}{r-1})^{4}}.$

Clearly, $\sum_{1\leq s\leq
r}\frac{W^{2}(H^{G}_{s})-W^{2}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}}\leq
0$. Thus for large $n$,

$(\rho\sum_{1\leq s\leq
r}\frac{W^{1}(H^{G}_{s})-W^{1}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}})+(\sum_{1\leq
s\leq
r}\frac{W^{2}(H^{G}_{s})-W^{2}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}})+\Theta(\frac{1}{\rho})<0.$

Recall that $\rho^{3}(f(G^{\prime},\rho)-f(G,\rho))\geq 0$, i.e.,

$(\rho\sum_{1\leq s\leq
r}\frac{W^{1}(H^{G}_{s})-W^{1}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}})+(\sum_{1\leq
s\leq
r}\frac{W^{2}(H^{G}_{s})-W^{2}(H^{G^{\prime}}_{s})}{(1+\frac{n_{s}}{\rho})^{2}})+\Theta(\frac{1}{\rho})\geq
0.$

This is a contradiction. Hence $G\in\mathcal{G}_{j}$ for some $1\leq j\leq r$
and $n_{j}=\lfloor\frac{n}{r}\rfloor$. Thus we obtain

${\rm SPEX}(\mathcal{G})=\cup_{i,n_{i}=\lfloor\frac{n}{r}\rfloor}{\rm
SPEX}(\mathcal{G}_{i}).$

For a graph $G^{\prime\prime}\in\mathcal{G}_{i}$ with $1\leq i\leq r$, note
that $H^{G^{\prime\prime}}=H^{G^{\prime\prime}}_{i}$. Since
$H^{G^{\prime\prime}}_{i}$ has $t$ edges, we can assume that the edges of
$H^{G^{\prime\prime}}_{i}$ are embedded into a fixed subset of size $2t$ in
$V_{i}$ (this fixed subset is viewed as $T$ in Theorem 1.2). By Theorem 1.2,
we see that $G^{\prime\prime}\in{\rm SPEX}(\mathcal{G}_{i})$ if and only if
$H^{G^{\prime\prime}}=H^{G^{\prime\prime}}_{i}\in{\rm
EX}^{\infty}(\mathcal{H}^{\mathcal{G}_{i}})$. This completes the proof .
$\Box$

Declaration of competing interest

There is no conflict of interest.

Data availability statement

No data was used for the research described in the article.

## References

  * [1] J. Byrne, D.N. Desai and M. Tait, A general theorem in spectral extremal graph theory, arXiv:2401.07266v1.
  * [2] D. Cvetković, P. Rowlinson and S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2010.
  * [3] S. Cioabă, L. Feng, M. Tait and X. Zhang, The maximum spectral radius of graphs without friendship subgraphs, Electron. J. Combin. 27 (2020) #4.22.
  * [4] S. Cioabă, D. Desai and M. Tait, The spectral even cycle problem, arXiv:2205.00990v1.
  * [5] S. Cioabă, D. Desai and M. Tait, The spectral radius of graphs with no odd wheels, European J. Combin. 99 (2022) 103420.
  * [6] S. Cioabă, D. Desai and M. Tait, A spectral Erdős-Sós theorem, SIAM J. Discrete Math. 37 (2023) 2228-2239.
  * [7] D. Desai, L. Kang, Y. Li, Z. Ni, M. Tait and J. Wang, Spectral extremal graphs for intersecting cliques, Linear Algebra Appl. 644 (2022) 234-258.
  * [8] L. Fang, H. Lin, J. Shu and Z. Zhang, Spectral extremal results on trees, arXiv:2401.05786v2.
  * [9] L. Fang, M. Tait and M. Zhai, Turán numbers for non-bipartite graphs and applications to spectral extremal problems, arXiv:2404.09069v1.
  * [10] D. Ismailescu and D. Stefanica, Minimizer graphs for a class of extremal problems, J. Graph Theory 39 (2002) 230-240.
  * [11] X. Lei and S. Li, Spectral extremal problem on disjoint color-critical graphs, Electron. J. Combin. 31 (2024) # 1.25.
  * [12] Y. Li, and Y. Peng, Refinement on spectral Turán’s theorem, SIAM J. Discrete Math. 37 (2023) 2462-2485.
  * [13] Y. Li, and Y. Peng, The spectral radius of graphs with no intersecting odd cycles, Discrete Math. 345 (2022) 112907.
  * [14] H. Lin, M. Zhai and Y. Zhao, Spectral radius, edge-disjoint cycles and cycles of the same length, Electron. J. Combin. 29 (2022) # 2.1.
  * [15] Z. Ni, J. Wang and L. Kang, Spectral extremal graphs for disjoint cliques, Electron. J. Combin. 30 (2023) #1.20.
  * [16] V. Nikiforov, The spectral radius of graphs without paths and cycles of specified length, Linear Algebra Appl. 432 (2010) 2243-2256.
  * [17] V. Nikiforov, Bounds on graph eigenvalues II, Linear Algebra Appl. 427 (2007) 183-189.
  * [18] J. Wang, L. Kang and Y. Xue, On a conjecture of spectral extremal problems, J. Combin. Theory Ser. B 159 (2023) 20-41.
  * [19] M. Zhai and H. Lin, Spectral extrema of graphs: Forbidden hexagon, Discrete Math. 343 (2020) 112028.
  * [20] W. Zhang, The spectral radius, maximum average degree and consecutive lengths of graphs, Graphs Combin. 40 (2024) 32.


# Many-body effects on the quasiparticle band structure and optical response
of single-layer penta-NiN2

Enesio Marinho Jr<EMAIL_ADDRESS>Departamento de Física e Química,
Universidade Estadual Paulista (UNESP),
Av. Brasil, 56, Ilha Solteira, 15385-007 São Paulo , Brazil  Instituto de
Física Teórica, Universidade Estadual Paulista (UNESP),
R. Dr. Bento Teobaldo Ferraz, 271, 01140-070 São Paulo, São Paulo, Brazil.
Cesar E. P. Villegas<EMAIL_ADDRESS>Departamento de Ciencias,
Universidad Privada del Norte, Lima 15434, Peru  Instituto de Física Teórica,
Universidade Estadual Paulista (UNESP),
R. Dr. Bento Teobaldo Ferraz, 271, 01140-070 São Paulo, São Paulo, Brazil.
Pedro Venezuela<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal Fluminense (UFF),
Av. Gal. Milton Tavares de Souza, s/n, 24210-346 Niterói, Rio de Janeiro,
Brazil.  Instituto de Física Teórica, Universidade Estadual Paulista (UNESP),
R. Dr. Bento Teobaldo Ferraz, 271, 01140-070 São Paulo, São Paulo, Brazil.
Alexandre R. Rocha<EMAIL_ADDRESS>Instituto de Física Teórica,
Universidade Estadual Paulista (UNESP),
R. Dr. Bento Teobaldo Ferraz, 271, 01140-070 São Paulo, São Paulo, Brazil.

###### Abstract

We present a comprehensive first-principles study on the optoelectronic
properties of the single-layer nickel diazenide (penta-NiN2), a recently
synthesized Cairo pentagonal 2D semiconductor. We carry out ab initio
calculations based on the density-functional theory (DFT) and many-body
perturbation theory, within the framework of Green’s functions, to describe
the quasiparticle properties and analyze the excitonic effects on the optical
properties of monolayer penta-NiN2. Our results reveal a quasiparticle band
gap of approximately 1 eV within the eigenvalue self-consistent $GW$ approach,
corroborating the monolayer penta-NiN2’s potential in optoelectronics.
Remarkably, the acoustic phonon-limited carrier mobility for the monolayer
penta-NiN2 exhibits an ultra-high hole mobility of $84{\times}10^{4}$
cm2/V$\cdot$s. Furthermore, our findings indicate that the material’s band gap
exhibits an anomalous negative dependence on temperature. Despite being a two-
dimensional material, monolayer penta-NiN2 presents resonant excitons in its
most prominent absorption peak. Therefore, penta-NiN2 boasts compelling and
promising properties that merit exploration in optoelectronics and high-speed
devices.

##  I Introduction

Motivated by the prediction of penta-graphene [1], a 2D carbon allotrope
composed entirely of pentagons, much effort has been devoted to discovering
novel pentagon-based 2D materials with intriguing properties. Since then, more
than a hundred pentagon-based 2D systems have been proposed [2], such as the
unitary sheets of penta-silicene [3, 4], penta-germanene [5], and penta-
tellurene [6], as well as the binary sheets penta-MS2 (M = Ni, Pd, Pt) [7],
penta-PdSe2 [8], among others [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19,
20].

Some of those pentagonal-based materials have already been synthesized.
Pentagonal silicene nanoribbons (NRs), for example, were synthesized by Cerdá
_et al._ [21], who grew these 1D structures on a Ag(110) surface.
Nevertheless, the first pentagon-based 2D sheet successfully synthesized was
the penta-PdSe2, which was obtained by exfoliating its bulk counterpart [22].
Combining physical and chemical vapor deposition techniques, Zhang _et al._
[23] reported the experimental realization of few-layered penta-PdS2.
Regarding ternary systems, the penta-PdPSe single-layer was successfully
fabricated by high-temperature solid-state reaction [24]. Also, Wang _et al._
[25] have synthesized penta-PdPS monolayer and few-layer via chemical vapor
transport methods.

Employing a combination of crystal-chemical design and high-pressure bottom-up
approach, Bykov _et al._ [26] have synthesized pentagonal nickel diazenide
(penta-NiN2), whose crystal structure is composed of distorted pentagonal
motifs forming an ideal prototype of Cairo tessellation (Fig. 1). Theoretical
calculations have predicted that this novel 2D pentagonal semiconductor
exhibits remarkably high elastic modulus, tensile strength, and room-
temperature lattice thermal conductivity [27]. Moreover, first-principles
calculations indicated that the single-layer penta-NiN2 is a direct band-gap
semiconductor, while the bulk phase is metallic [26]. Thus, at the PBE-DFT
level of theory, its electronic band gap yielded about $0.05$ eV [28, 27],
whereas at the hybrid functional HSE06 level, its band gap increases to $1.1$
eV, suggesting that the monolayer penta-NiN2 holds potential for
optoelectronic applications [28, 27, 26]. Despite this, experimental
measurements of the fundamental band gap for single-layer penta-NiN2 are still
lacking. Hence, accurate predictions of its electronic and optical properties
based on many-body perturbation theory (MBPT) can provide useful insights.

Figure 1: (Color online) (a) Top and (b) side views of the crystal structure
of monolayer penta-NiN2, with Ni and N atoms being represented by grey and
blue spheres, respectively. The square unit cell is represented by black solid
lines.

Herewith, we analyze the role of the electron-electron and electron-hole
interactions in the electronic band gap renormalization and the optical
properties of penta-NiN2. Our quasiparticle band structure within eigenvalue
self-consistent $GW$ indicates that the monolayer penta-NiN2 is a direct band
gap semiconductor, with a band gap magnitude of $1.05$ eV. Moreover, our
results indicate that this pentagonal 2D semiconductor presents ultrahigh hole
mobility, and a negative pressure coefficient which suggests an anomalous
temperature dependence of the band gap. Finally, optical absorption has shown
that excitonic effects play an important role in its spectrum, with the
presence of dark excitons and bright resonant states.

Figure 2: (Color online) Electronic band structure of monolayer penta-NiN2
computed within (a) HSE06, (b) $G_{0}W_{0}$ and (c) ev$GW$. The DFT-PBE
results (dashed black lines) are shown for comparison. The energies refer to
the valence band maximum.

##  II Computational Methods

The ground-state structural and electronic properties have been obtained by
means of density-functional theory (DFT) [29, 30], as implemented in the
quantum espresso package [31]. The Perdew-Burke-Ernzerhof generalized-gradient
approximation (GGA-PBE) [32] and the screened hybrid functional of Heyd,
Scuseria, and Ernzerhof (HSE06) [33, 34] were employed to describe the
exchange-correlation functional. Fully relativistic ONCVPSP [35] norm-
conserving pseudopotentials are used to describe the electron–ion interaction.
Equilibrium structural properties have been computed by fully relaxing both
the in-plane unit cell and the atomic positions, with a force convergence
criterion of $1{\times}10^{-5}$ Ry/Bohr. A vacuum spacing of ${\sim}16$ Å was
used in the out-of-plane direction to avoid spurious interaction with the
periodic image of the system. A kinetic energy cutoff of 96 Ry was adopted to
expand the Kohn-Sham orbitals in a plane-wave basis set. The charge density
has been calculated by sampling the Brillouin zone with a Monkhorst-Pack
$\Gamma$-centered _k_ -mesh of $14{\times}14{\times}1$.

Quasiparticle energies are calculated as corrections to the Kohn–Sham
eigenvalues using the Green’s function formulation of MBPT. Taking the first
order Taylor expansion of the self-energy $\Sigma_{n\mathbf{k}}$ around the
Kohn-Sham eigenvalues $\epsilon_{n\mathbf{k}}^{\text{KS}}$ (Newton
approximation), we have

$E_{n\mathbf{k}}^{\text{QP}}=\epsilon_{n\mathbf{k}}^{\text{KS}}+Z_{n\mathbf{k}}\matrixelement{n\mathbf{k}^{\text{KS}}}{\Sigma_{n\mathbf{k}}(\epsilon_{n\mathbf{k}}^{\text{KS}})-V^{\text{xc}}_{n\mathbf{k}}}{n\mathbf{k}^{\text{KS}}}\,,$
(1)

where
$Z_{n\mathbf{k}}=\left[1-\evaluated{\derivative*{\Sigma_{n\mathbf{k}}(\omega)}{\omega}}_{\omega=\epsilon_{n\mathbf{k}}^{\text{KS}}}\right]^{-1}$
is the quasiparticle renormalization factor, $\ket{n\mathbf{k}^{\text{KS}}}$
describes the Kohn-Sham eigenstates, $V^{\text{xc}}_{n\mathbf{k}}$ is the DFT
exchange-correlation potential. In the $G_{0}W_{0}$ approximation [36], the
self-energy is described by $\Sigma=iG_{0}W_{0}$, corresponding to the first
iteration of Hedin’s equations [37]. The yambo code [38, 39] was used to
perform these calculations. We employ an energy cutoff of 100 Ry for the
exchange part of the self-energy. For the correlation part of the self-energy,
the dielectric screening matrix is constructed considering the Plasmon-Pole
approximation [40] with a $\mathbf{G}$-vector energy cutoff of 24 Ry and
including 800 bands. The GW sum-over-states is performed on 1800 bands. This
number of bands is sufficient to provide accurate results owing to the
implementation of the Bruneval-Gonze terminators approach [41].

To reduce spurious interactions among periodic images in the out-of-plane
direction, we described the Coulomb potential as a truncated potential in a
slab geometry with a cutoff of 8 Ry. Furthermore, to speed up the convergence
of the $GW$ calculations w.r.t the $k$-point sampling, we employ the W-av
method [42], which combines Monte Carlo integration techniques with an
interpolation scheme of the screened potential. We have set the
$\mathbf{G}$-vectors cutoff to 1.2 Ry. According to our convergence tests (see
Supplementary Information), adopting the described parameters one can compute
the quasiparticle band gap of the monolayer penta-NiN2 within a convergence
threshold of $0.1$ eV.

To overcome some issues associated with the rather pronounced starting-point
dependence of the $G_{0}W_{0}$ approach, mainly regarding semi-local
functional-based narrow band gap semiconductors, we also compute the
quasiparticle energies within the eigenvalue self-consistent $GW$ (ev$GW$). In
this approach, the input Kohn-Sham eigenvalues are replaced by the
quasiparticle energies from the previous iteration, and the updated
eigenvalues are used to construct $G$ and $W$, until the self-consistency in
the quasiparticle energies is reached. Our convergence analysis for the ev$GW$
calculations indicated that self-consistency has been achieved after 3
iterations.

The excitonic effects on optical spectra are analyzed by solving the Bethe-
Salpeter equation (BSE) in the resonant (Tamm-Dancoff) approximation [43, 44],
also employing the yambo code [38, 39]. The BSE Hamiltonian is given by

$H^{\text{BSE}}_{\begin{subarray}{c}vc\mathbf{k}\\\
v^{\prime}c^{\prime}\mathbf{k}^{\prime}\end{subarray}}=\left(\epsilon^{\text{QP}}_{c\mathbf{k}}-\epsilon^{\text{QP}}_{v\mathbf{k}}\right)\variation_{c,c^{\prime}}\variation_{v,v^{\prime}}\variation_{\mathbf{k},\mathbf{k}^{\prime}}+\Xi^{\text{eh}}_{vc\mathbf{k},v^{\prime}c^{\prime}\mathbf{k}^{\prime}}\,,$
(2)

where the first term in parenthesis comprises a diagonal part that contains
the quasiparticle energy differences, and $\Xi^{\text{eh}}=K^{x}+K^{c}$ is the
so-called BSE kernel, composed of an $e$-$h$ attraction term ($K^{c}$) and a
repulsive exchange ($K^{x}$)[45].

Thus, by diagonalizing the BSE Hamiltonian,

$\sum_{v^{\prime}c^{\prime}\mathbf{k}^{\prime}}H^{\text{BSE}}_{\begin{subarray}{c}vc\mathbf{k}\\\
v^{\prime}c^{\prime}\mathbf{k}^{\prime}\end{subarray}}A^{S}_{v^{\prime}c^{\prime}\mathbf{k}^{\prime}}=\Omega_{S}A^{S}_{vc\mathbf{k}}\,,$
(3)

we obtain the exciton envelope function $A^{S}_{vc\mathbf{k}}$ and its
eigenvalues $\Omega_{S}$, for each exciton $S$.

Converged optical absorption spectra have been obtained including the three
highest-occupied valence bands and the three lowest-unoccupied conduction
bands, with a fine $\mathbf{k}$-grid sampling of $40{\times}40{\times}1$.

The optical response can be analyzed by the macroscopic dielectric function,
which is computed using the exciton eigenstates and eigenenergies as follows

$\displaystyle\epsilon_{M}(\omega)$ $\displaystyle=$ $\displaystyle
1-\lim_{\mathbf{q}\to 0}\frac{8\pi
e^{2}}{\absolutevalue{\mathbf{q}}^{2}VN_{q}}\sum_{\begin{subarray}{c}vc\mathbf{k}\\\
v^{\prime}c^{\prime}\mathbf{k}^{\prime}\end{subarray}}\rho^{*}_{vc\mathbf{k}}(\mathbf{q},\mathbf{G})\rho_{v^{\prime}c^{\prime}\mathbf{k}^{\prime}}(\mathbf{q},\mathbf{G}^{\prime})$
(4) $\displaystyle\times$
$\displaystyle\sum_{S}\frac{A^{S}_{vc\mathbf{k}}\left(A^{S}_{v^{\prime}c^{\prime}\mathbf{k}^{\prime}}\right)^{*}}{\omega-\Omega_{S}}$

where
$\rho_{vc\mathbf{k}}(\mathbf{q},\mathbf{G})=\matrixelement{v\mathbf{k}}{e^{i(\mathbf{q}+\mathbf{G})\cdot\mathbf{r}}}{c\mathbf{k}-\mathbf{q}}$
are the single-particle dipole matrix elements, $\mathbf{G}$ is the reciprocal
lattice vector, $N_{q}$ is number of transferred momenta $\mathbf{q}$, $V$ is
the real-space unit cell volume.

However, it is noteworthy to stress that when employing the Coulomb truncation
scheme, the real and imaginary part of the macroscopic dielectric function
goes to unity and zero, respectively. This is a consequence of simulating an
atom-thick structure that is fully surrounded by a vacuum. In this regard, the
imaginary part of the polarizability per unit area ($\alpha_{2}$) is related
to the optical absorbance [46, 47]

$A(\omega)=\frac{4\pi\omega}{c}\alpha_{2}(\omega)\,.$ (5)

Hereafter, the optical absorption spectrum of the monolayer penta-NiN2 will be
described using the absorbance as the descriptive observable.

##  III Results and Discussion

The relaxed crystal structure of penta-NiN2 is shown in Fig. 1. It consists of
an atomic-thick planar sheet composed of Ni2N3 pentagons, with tetragonal
P$4/mbm$ symmetry (space group no. 127). Our calculated lattice parameters are
$a=b=4.53$ Å, and the bond lengths of the Ni-N and N-N bonds are $1.875$ Å and
$1.246$ Å, respectively. These values reproduce the experimental structural
parameters [26] with errors below 1.5%, and are also consistent with previous
theoretical results [28, 48].

The electronic band structure of penta-NiN2 was first computed at the DFT-PBE
level. We obtained a direct bandgap of 0.07 eV, which agrees with prior
theoretical results that indicate a PBE band gap smaller than 0.1 eV [28, 48].
We also calculated the electronic band gap using the hybrid HSE06 functional,
obtaining a magnitude of 1.14 eV. This result is also in close agreement with
previously reported values of $\sim 1.1$ eV [28, 48].

We further proceeded to compute the quasiparticle band structures of penta-
NiN2 at both $G_{0}W_{0}$ and ev$GW$ levels, as shown in Fig. 2. The obtained
$G_{0}W_{0}$ bandgap was 0.65 eV, whereas the quasiparticle band gap within
ev$GW$ yielded 1.05 eV. The PBE-DFT band structure is also shown for
comparison.

In general, the MBPT formalism provides a suitable and accurate framework to
incorporate the electronic correlation beyond the single-particle mean-field
theories. However, the $G_{0}W_{0}$ results exhibit a dependence on the
description of the starting Kohn-Sham eigensystem. To clarify this starting-
point dependence, we also compare the monolayer penta-NiN2 results to the
reported results for germanium [49].

The GGA-PBE band gap of the germanium is 0.08 eV, substantially underestimated
and with the same magnitude that we obtained for the penta-NiN2 at this level
of theory. In addition, the quasiparticle band gap of the germanium computed
within $G_{0}W_{0}$ approach is still underestimated, yielding 0.47 eV. Thus,
other $GW$ flavors should be employed. The eigenvalue self-consistent $GW$
(ev$GW$) is noticeably less dependent on the DFT starting point, providing a
higher quasiparticle band gap of 0.95 eV , which is expected owing to the
self-consistent update of the quasiparticle corrections.

Figure 3: Electronic band gap as a function of the pressure for the monolayer
penta-NiN2. We apply the quasiparticle correction within ev$GW$ approach as a
scissor operator $\Delta_{\text{QP}}=0.97$ eV to the PBE-DFT band gaps.

As a result, we expect that the experimental band gap of the penta-NiN2 should
be about 1 eV, close to the result yielded within the ev$GW$ approach,
indicating that this pentagon-based 2D semiconductor is indeed a promising
candidate for optoelectronics.

Figure 4: (Color online) Optical properties of the monolayer penta-NiN2. (a)
Absorbance along [100] and [010] directions including excitonic effects by
solving the Bethe-Salpeter equation (BSE) and at the independent particle (IP)
level. (b) Oscillator strengths of the optical transitions normalized to 1.
Exciton wavefunctions (top panels) and exciton weights on the Brillouin zone
(bottom panels) for (c) the first dark exciton (XD) and (d) the most intense
bright exciton (XA). The isosurface in orange ($0.005$ e/Å3) represents the
electron density associated with the hole at a fixed N site (white spheres).

We turn to analyze the carrier mobilities in single-layer penta-NiN2. For 2D
systems with isotropic effective mass [50], the intrinsic mobility limited by
acoustic-phonon scattering can be estimated through Bardeen–Shockley
deformation potential theory [51] as follows

$\mu_{2\text{D}}=\frac{e\hbar^{3}C_{\text{2D}}}{k_{B}T\absolutevalue{m_{i}^{*}}^{2}\left(\Xi_{1}\right)^{2}}\,,$
(6)

where $C_{\text{2D}}$ is the longitudinal elastic constant, extracted from the
relation $(E-E_{0})/S_{0}=C_{2\text{D}}\delta^{2}/2$, with $E$ being the total
energy for the strained system, $E_{0}$ and $S_{0}$ the total energy and the
surface area for the system at equilibrium, and $\delta=\Delta l/l_{0}$ is the
applied uniaxial strain. $\Xi_{1}$ is the acoustic deformation potential
defined as $\Xi_{1}=\Delta E_{\text{b}}/\delta$, where $\Delta E_{\text{b}}$
describes the energy shifts in the valence or conduction band edge, under
proper compressive or tensile strain. We have computed those features applying
a uniaxial strain $\delta$ varying from $-2\%$ to $2\%$, in steps of $0.5\%$,
along the direction [110]. The calculated carrier effective masses,
longitudinal elastic constant, and mobilities are summarized in Table 1. The
obtained longitudinal elastic constant, $C_{\text{2D}}=175.9$ N/m, is in close
agreement with other reported theoretical results of $172.2$ N/m [52] and
174.9 N/m [28] .

Table 1:  Predicted carrier mobilities of monolayer penta-NiN2 and the relevant parameters [see Eq. (6)] along the [110] direction. The results have been calculated at $T=300$ K, within the DFT-PBE level.  Carrier |  $m^{*}$ ($m_{0}$) |  $C_{\text{2D}}$ (N/m) |  $\Xi_{1}$ (eV) |  $\mu$ $\left(\times 10^{4}\text{ cm}^{2}/\text{V$\cdot$ s}\right)$
---|---|---|---|---
$e$ | 0.34 | 175.9 | 2.72 | 0.44
$h$ | 0.097 | 175.9 | 0.69 | 83.6

The computed mobilities yielded ultra-high hole mobility of ${\sim}84\times
10^{4}$ cm2/V$\cdot$s, which is one or two orders of magnitude higher than
that of electrons, owing to the lowest effective mass and deformation
potential for holes. These results are similar to the carriers mobilities of
other pentagonal-based semiconductors, such as penta-PdP2, with $\mu_{h}\sim
84\times 10^{4}$ cm2/V$\cdot$s, and penta-PdAs2, with $\mu_{h}\sim 89\times
10^{4}$ cm2/V$\cdot$s [53]. Again for those materials, the electron mobilities
are two orders of magnitude smaller than the hole mobilities. The remarkably
high carrier mobility of monolayer penta-NiN2 holds great potential for high-
speed electronic devices.

In addition, we analyzed the pressure dependence of the band gap for monolayer
penta-NiN2. The effect on the gap due to the contraction of the lattice with
decreasing temperature is intimately related to the response of the electronic
band structure upon applying an external hydrostatic pressure [54]. The
thermal expansion effects (TE) on the band gap can be estimated by

$\left(\partialderivative{E_{g}}{T}\right)_{\text{TE}}=-\alpha_{V}B_{0}\derivative{E_{g}}{P}\,,$
(7)

where $-\alpha_{V}$ is the volumetric expansion coefficient, $B_{0}$ is the
bulk modulus, and $\differential E/\differential P$ is the so-called pressure
coefficient of the gap [54]. At room temperature, $\alpha_{V}$ is positive,
and the sign of Eq. (7) for the thermal expansion contribution depends on the
sign of the pressure coefficient. For most direct bandgap semiconductors, the
pressure coefficient is positive, and therefore a thermal expansion causes a
gap reduction [54]. For instance, the pressure coefficient for GaAs was
experimentally estimated and it yields a positive value of $\sim 80-130$
meV/GPa [55].

The pressure coefficient can be used to quantitatively estimate the
temperature dependence of the band gap of semiconductors, such as black
phosphorus [56] and lead halide perovskites [54]. Thus, for penta-NiN2, we
plot the band gap as a function of the crystal pressure, as shown in Fig. 3.
As a result, we compute a negative pressure coefficient of $-49$ meV/GPa,
indicating that this pentagonal 2D semiconductor exhibits an anomalous
temperature dependence, which means that its bandgap tends to increase with
increasing temperature.

Figure 5: (Color online) Orbital-resolved bandstructures of penta-NiN2
projected into Ni(d) and N(p) orbitals. The color scale indicates the
magnitude of the projection. The Kohn-Sham eigenvalues were shifted by a
scissor operator of 0.97 eV, corresponding to the computed quasiparticle
correction within the eigenvalue self-consistent $GW$ (ev$GW$). (b) The scheme
of the main optical transitions is depicted, with a representation of the dark
(XD) and the most intense resonant exciton (XA) at the M valley in reciprocal
space.

The optical spectrum of the monolayer penta-NiN2 including excitonic effects
is shown in Fig. 4(a). Those calculations were carried out by solving the BSE
with $evGW$ quasiparticle corrections for the band gap. The calculated
absorption spectra were also obtained at the independent particle (IP) level,
for light-polarized in [100] and [010] directions. The most prominent feature
is the isotropic optical absorbance yielded from the centrosymmetric structure
of penta-NiN2 at both BSE and IP level of theory. As expected, when the
electron-hole interaction is switched on, it induces a red shift and a
redistribution of the oscillator strengths.

An optically forbidden (allowed) dark (bright) excitonic transition can be
described according to the very small (high) oscillator strength of optical
transitions. The normalized oscillator strengths for the monolayer penta-NiN2
are shown in Fig. 4(b). The ground-state exciton XD is dark, with a binding
energy of 0.34 eV. The most prominent excitonic peak is located at $\sim 1.1$
eV, slightly above the quasiparticle band gap within the ev$GW$ approach.
Therefore, the exciton XA exciton can be characterized as a resonant exciton,
which occurs due to the presence of strong many-body interactions [57].

The presence of resonant excitons has been theoretically predicted for
graphene [58], and it has already been experimentally observed in both
graphene [59] and WSe2 monolayer [57]. For the latter, the resonant excitons
were identified by combining temperature-dependent high-resolution
spectroscopic ellipsometry at low temperatures with high-energy
photoluminescence spectroscopy. In contrast to Wannier-Mott excitons, the
free-carrier nature of resonant excitons offers potential advantages for
optoelectronics, once the separation of the photoexcited carriers is relevant
[60].

To gain further insight into the optical spectra, we plot the square modulus
of the wave function for the ground-state dark exciton XD and the most
prominent bright exciton XA, as shown in Figs. 4(c) and 4(d), respectively. To
obtain these isosurfaces, we fix the hole at a N site and then project the
exciton probability amplitude onto the $x{-}y$ plane. We notice that the main
and secondary diagonals play important roles in the real-space electron
distribution, due to the position of the atoms and the directions of the
nitrogen-nitrogen bonds. Regarding diagonal directions, we can verify exciton
resonant states for both XD and XA, so that they have characteristic residual
amplitudes that are not significantly decaying away from the hole [61]. All
the main optical transitions occur in the vicinity of the $M$ point, and the
exciton XA is due to transitions from the valence band maximum (VBM) to the
third conduction band (CBM+2).

The selection rules involved in the optical transitions for monolayer penta-
NiN2 were investigated by means of the orbital-resolved band structure. We
apply a scissor operator of $0.97$ eV in the PBE band structure, which
corresponds to the calculated ev$GW$ quasiparticle gap renormalization. The
results for Ni(_d_) and N(_p_) orbitals are shown in Fig. 5(a).

As we can verify, the VBM and CBM originated from the _p_ orbitals of N atoms,
whereas the CBM+2 is composed of _d_ orbitals from Ni atoms. Since the most
prominent bright exciton XA occurs due to optical transitions from VBM to
CBM+2 (see Fig. 4(d), bottom panel), we can state that the Laporte or orbital
selection rule governs the optical transitions in single-layer penta-NiN2.
Taking into account that its structure is centrosymmetric, the Laporte
selection rule establishes that electronic transitions that conserve the
wavefunction parity are forbidden, implying that _p_ to _p_ , or _d_ to _d_ ,
transitions should not be observed. The _p_ to _d_ optical transition verified
for exciton XA is orbitally allowed with a change in the azimuthal quantum
number $\Delta\ell=+1$. On the other hand, transitions from VBM to CBM tend to
be forbidden since $\Delta\ell=0$. The dark and bright excitons in single-
layer penta-NiN2 considering the optical transitions either allowed or
forbidden by Laporte rule are summarized in Fig. 5(b).

##  IV Conclusions

In conclusion, we investigated the optoelectronic properties of monolayer
penta-NiN2, a recently synthesized pentagonal 2D semiconductor with promising
performance in optoelectronic devices. Employing state-of-the-art parameter-
free excited-state calculations, within the $GW$ approach, and by solving the
Bethe-Salpeter equation, we have computed the quasiparticle band structure and
the optical absorption spectrum including excitonic effects. Employing the
eigenvalue self-consistent $GW$ approach, its quasiparticle band gap yielded
${\sim}1$ eV. The ultra-high hole mobility of ${\sim}80\times 10^{4}$
cm2/V$\cdot$s and the negative pressure coefficient for this pentagonal 2D
semiconductor contribute to a set of promising potential for high-speed
electronic devices. The optical absorption spectrum indicates that the ground-
state exciton XD is dark, while the most intense optical peak XA is a resonant
exciton with a free-carrier nature. The optical transitions in monolayer
penta-NiN2 can be explained by means of the Laporte selection rule, providing
a comprehensive description of the absorption features and excitonic states in
this 2D pentagonal semiconductor. Therefore, we shed light on promising
optoelectronic properties of single-layer penta-NiN2, paving the way for
potential applications using this pentagonal 2D semiconductor.

## Acknowledgements

E.M.Jr. acknowledges the financial support from the Brazilian agency FAPESP,
Grant No. 20/13172-8 and 2017/02317-2. This research was supported by
resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of
the UNESP and the Centro Nacional de Processamento de Alto Desempenho em São
Paulo (CENAPAD-SP).

## References

  * Zhang  _et al._ [2015] S. Zhang, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe, and P. Jena, Proc. Natl. Acad. Sci. 112, 2372 (2015).
  * Shen and Wang [2022] Y. Shen and Q. Wang, Phys. Rep. 964, 1 (2022).
  * Ding and Wang [2015] Y. Ding and Y. Wang, J. Mater. Chem. C 3, 11341 (2015).
  * Guo  _et al._ [2019] Y. Guo, C. Zhang, J. Zhou, Q. Wang, and P. Jena, Phys. Rev. Appl. 11, 064063 (2019).
  * Gao  _et al._ [2019] Z. Gao, Z. Zhang, G. Liu, and J.-S. Wang, Phys. Chem. Chem. Phys. 21, 26033 (2019).
  * Zhang  _et al._ [2021a] T. Zhang, J.-H. Lin, X.-L. Zhou, and X. Jia, Appl. Surf. Sci. 559, 149851 (2021a).
  * Yang  _et al._ [2020] H. Yang, Y. Li, Z. Yang, X. Shi, Z. Lin, R. Guo, L. Xu, H. Qu, and S. Zhang, Vacuum 174, 109176 (2020).
  * Oyedele  _et al._ [2017a] A. D. Oyedele, S. Yang, L. Liang, A. A. Puretzky, K. Wang, J. Zhang, P. Yu, P. R. Pudasaini, A. W. Ghosh, Z. Liu, C. M. Rouleau, B. G. Sumpter, M. F. Chisholm, W. Zhou, P. D. Rack, D. B. Geohegan, and K. Xiao, J. Am. Chem. Soc. 139, 14090 (2017a).
  * Zhang  _et al._ [2016] S. Zhang, J. Zhou, Q. Wang, and P. Jena, J. Phys. Chem. C 120, 3993 (2016).
  * Li  _et al._ [2015] F. Li, K. Tu, H. Zhang, and Z. Chen, Phys. Chem. Chem. Phys. 17, 24151 (2015).
  * Li  _et al._ [2016] J. Li, X. Fan, Y. Wei, and G. Chen, Sci. Rep. 6, 1 (2016).
  * Liu  _et al._ [2016] H. Liu, G. Qin, Y. Lin, and M. Hu, Nano Lett. 16, 3831 (2016).
  * Zhao  _et al._ [2020] K. Zhao, Y. Guo, Y. Shen, Q. Wang, Y. Kawazoe, and P. Jena, J. Phys. Chem. Lett. 11, 3501 (2020).
  * Hou  _et al._ [2022] C. Hou, Y. Shen, W. Sun, Y. Chen, D. Ni, and Q. Wang, J. Mater. Chem. C 10, 10302 (2022).
  * Sun  _et al._ [2021] W. Sun, Y. Shen, Y. Guo, Y. Chen, and Q. Wang, Appl. Surf. Sci. 554, 149499 (2021).
  * Jing  _et al._ [2017] Y. Jing, Y. Ma, Y. Wang, Y. Li, and T. Heine, Chem. Eur. J. 23, 13612 (2017).
  * Iqbal  _et al._ [2023] T. Iqbal, G. Rehman, I. Khan, R. Ahmad, and I. Ahmad, J. Phys. Chem. Solids 177, 111280 (2023).
  * Liu and Zhuang [2019] L. Liu and H. L. Zhuang, Comput. Mater. Sci. 166, 105 (2019).
  * Bravo  _et al._ [2021] S. Bravo, M. Pacheco, V. Nuñez, J. D. Correa, and L. Chico, Nanoscale 13, 6117 (2021).
  * Liu  _et al._ [2020] X. Liu, X. Shao, Y. Zheng, S. Qi, M. Zhang, H. Li, C. Li, M. Tan, D. Zheng, and M. Zhao, Appl. Surf. Sci. 518, 146197 (2020).
  * Cerdá  _et al._ [2016] J. I. Cerdá, J. Sławińska, G. Le Lay, A. C. Marele, J. M. Gómez-Rodríguez, and M. E. Dávila, Nat. Comm. 7, 13076 (2016).
  * Oyedele  _et al._ [2017b] A. D. Oyedele, S. Yang, L. Liang, A. A. Puretzky, K. Wang, J. Zhang, P. Yu, P. R. Pudasaini, A. W. Ghosh, Z. Liu, C. M. Rouleau, B. G. Sumpter, M. F. Chisholm, W. Zhou, P. D. Rack, D. B. Geohegan, and K. Xiao, J. Am. Chem. Soc. 139, 14090 (2017b).
  * Zhang  _et al._ [2021b] X. Zhang, G. Su, J. Lu, W. Yang, W. Zhuang, K. Han, X. Wang, Y. Wan, X. Yu, and P. Yang, ACS Appl. Mater. Interfaces 13, 43063 (2021b).
  * Li  _et al._ [2021] P. Li, J. Zhang, C. Zhu, W. Shen, C. Hu, W. Fu, L. Yan, L. Zhou, L. Zheng, H. Lei, Z. Liu, W. Zhao, P. Gao, P. Yu, and G. Yang, Advanced Materials 33, 2102541 (2021).
  * Wang  _et al._ [2022] X. Wang, T. Xiong, K. Zhao, Z. Zhou, K. Xin, H.-X. Deng, J. Kang, J. Yang, Y.-Y. Liu, and Z. Wei, Adv. Mater. 34, 2107206 (2022).
  * Bykov  _et al._ [2021] M. Bykov, E. Bykova, A. V. Ponomareva, F. Tasnádi, S. Chariton, V. B. Prakapenka, K. Glazyrin, J. S. Smith, M. F. Mahmood, I. A. Abrikosov, and A. F. Goncharov, ACS Nano 15, 13539 (2021).
  * Mortazavi  _et al._ [2022] B. Mortazavi, X. Zhuang, T. Rabczuk, and A. V. Shapeev, Phys. E: Low-Dimens. Syst. Nanostructures. 140, 115221 (2022).
  * Yuan  _et al._ [2019a] J.-H. Yuan, Y.-Q. Song, Q. Chen, K.-H. Xue, and X.-S. Miao, Appl. Surf. Sci. 469, 456 (2019a).
  * Hohenberg and Kohn [1964] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  * Kohn and Sham [1965] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  * Giannozzi  _et al._ [2009] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, _et al._ , J. Phys.: Condens. Matter 21, 395502 (2009).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Heyd  _et al._ [2003] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).
  * Heyd  _et al._ [2006] J. Heyd, G. E. Scuseria, and M. Ernzerhof, TJ. Chem. Phys. 124, 219906 (2006).
  * Hamann [2013] D. R. Hamann, Phys. Rev. B 88, 085117 (2013).
  * Hedin [1965] L. Hedin, Phys. Rev. 139, A796 (1965).
  * Golze  _et al._ [2019] D. Golze, M. Dvorak, and P. Rinke, Front. Chem. 7, 377 (2019).
  * Marini  _et al._ [2009] A. Marini, C. Hogan, M. Grüning, and D. Varsano, Comput. Phys. Commun. 180, 1392 (2009).
  * Sangalli  _et al._ [2019] D. Sangalli, A. Ferretti, H. Miranda, C. Attaccalite, I. Marri, E. Cannuccia, P. Melo, M. Marsili, F. Paleari, A. Marrazzo, G. Prandini, P. Bonfà, M. O. Atambo, F. Affinito, M. Palummo, A. Molina-Sánchez, C. Hogan, M. Grüning, D. Varsano, and A. Marini, J. Phys.: Cond. Matter 31, 325902 (2019).
  * Godby and Needs [1989] R. W. Godby and R. J. Needs, Phys. Rev. Lett. 62, 1169 (1989).
  * Bruneval and Gonze [2008] F. Bruneval and X. Gonze, Phys. Rev. B 78, 085125 (2008).
  * Guandalini  _et al._ [2023] A. Guandalini, P. D’Amico, A. Ferretti, and D. Varsano, npj Comput. Mater. 9, 44 (2023).
  * Tamm [1945] I. Tamm, J. Phys. (USSR) 9, 449 (1945).
  * Dancoff [1950] S. M. Dancoff, Phys. Rev. 78, 382 (1950).
  * Rohlfing and Louie [2000] M. Rohlfing and S. G. Louie, Phys. Rev. B 62, 4927 (2000).
  * Bechstedt  _et al._ [2012] F. Bechstedt, L. Matthes, P. Gori, and O. Pulci, Appl. Phys. Lett. 100, 261906 (2012).
  * Bernardi  _et al._ [2013] M. Bernardi, M. Palummo, and J. C. Grossman, Nano Lett. 13, 3664 (2013).
  * Zhang  _et al._ [2022a] C. Zhang, J. Sun, Y. Shen, W. Kang, and Q. Wang, J. Phys. Chem. Lett. 13, 5734 (2022a), pMID: 35713616.
  * Gant  _et al._ [2022] S. E. Gant, J. B. Haber, M. R. Filip, F. Sagredo, D. Wing, G. Ohad, L. Kronik, and J. B. Neaton, Phys. Rev. Mater. 6, 053802 (2022).
  * Lang  _et al._ [2016] H. Lang, S. Zhang, and Z. Liu, Phys. Rev. B 94, 235306 (2016).
  * Bardeen and Shockley [1950] J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950).
  * Zhang  _et al._ [2022b] C. Zhang, J. Sun, Y. Shen, W. Kang, and Q. Wang, J. Phys. Chem. Lett. 13, 5734 (2022b).
  * Yuan  _et al._ [2019b] J.-H. Yuan, B. Zhang, Y.-Q. Song, J.-F. Wang, K.-H. Xue, and X.-S. Miao, J. Mater. Sci. 54, 7035 (2019b).
  * Francisco-López  _et al._ [2019] A. Francisco-López, B. Charles, O. J. Weber, M. I. Alonso, M. Garriga, M. Campoy-Quiles, M. T. Weller, and A. R. Goñi, J. Phys. Chem. Lett. 10, 2971 (2019).
  * Chang  _et al._ [1984] K.-J. Chang, S. Froyen, and M. L. Cohen, Solid State Commun. 50, 105 (1984).
  * Villegas  _et al._ [2016] C. E. P. Villegas, A. Rocha, and A. Marini, Nano Lett. 16, 5095 (2016).
  * Tang  _et al._ [2019] C. S. Tang, X. Yin, M. Yang, D. Wu, M. D. Birowosuto, J. Wu, C. Li, C. Hettiarachchi, X. Y. Chin, Y.-H. Chang, _et al._ , ACS Nano 13, 14529 (2019).
  * Yang  _et al._ [2009] L. Yang, J. Deslippe, C.-H. Park, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 103, 186802 (2009).
  * Mak  _et al._ [2011] K. F. Mak, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 106, 046401 (2011).
  * Palummo  _et al._ [2023] M. Palummo, M. Re Fiorentin, K. Yamashita, I. E. Castelli, and G. Giorgi, J. Phys. Chem. Lett. 14, 1548 (2023).
  * Liang  _et al._ [2012] Y. Liang, S. Huang, and L. Yang, J. Mater. Res. 27, 403 (2012).


Subsection 3.6: $\ell(\lambda)$, $\mathscr{P}^{(d)}$, $\mathscr{D}^{(d)}$,
$V_{\lambda}$, $\Lambda_{d,r}$, $|\lambda|$.

Subsection 3.7: $\mathcal{C}_{L}$, $\mathcal{C}_{R}$, $\mathcal{R}_{L}$,
$\underline{\mathcal{R}}_{L}$, $\mathcal{R}_{R}$,
$\underline{\mathcal{R}}_{R}$, $\check{\mathcal{R}}_{L}$,
$\underline{\check{\mathcal{R}}}_{L}$, $\check{\mathcal{R}}_{R}$,
$\underline{\check{\mathcal{R}}}_{R}$, $\mathcal{R}_{LR}$,
$\underline{\mathcal{R}}_{LR}$, $\check{\mathcal{R}}_{LR}$,
$\underline{\check{\mathcal{R}}}_{LR}$, $\mathscr{PD}^{\mathrm{gr}}$,
$\mathscr{PD}$.

Subsection 3.9: $\mathscr{A}_{k,l,n}$, $\mathscr{A}_{k,l,n}^{\mathrm{gr}}$,
$\boldsymbol{\kappa}_{r,n}$, $(\boldsymbol{e}^{\mathrm{gr}})_{m^{\prime}\times
n^{\prime}}^{m\times n}$.

Subsection 3.10: $\phi_{PD}$, $X\cdot f$.

Subsection 3.11: $\mathscr{P}^{(\leq k)}$, $\prec$, $\mathbf{c}(a)$.

Subsection 3.13: $\mathsf{P}$, $\mathsf{P}_{k,l,n}$,
$\mathscr{PD}^{\mathrm{gr},(r,s)}$, $\mathscr{A}_{k,l,n}^{\mathrm{gr},(r,s)}$,

Subsection 3.14: $\mathring{U}_{L}$, $\mathring{U}_{R}$, $\mathring{U}_{LR}$.

Subsection 3.15: $\eta_{m,n}$.

Subsection 3.17: $\lhd$, $\mathbf{c}(a,b)$, $t^{\mathsf{a}}$,
$\partial^{\mathsf{a}}$.

Subsection 4.1: $\lambda_{L,a}$, $\lambda_{R,b}$, $K_{\lambda_{L,a}}$,
$K_{\lambda_{R,b}}$

Subsection 6.1: $\mathscr{PD}_{(\epsilon_{L})}$,
$\mathscr{PD}_{(\epsilon_{R})}$, $\epsilon_{L}$, $\epsilon_{R}$,
$\left(\mathscr{A}_{k,l,n}\right)_{(\epsilon_{R})}$,
$\left(\mathscr{A}^{\mathrm{gr}}_{k,l,n}\right)_{(\epsilon_{R})}$,
$\left(\mathscr{A}_{k,l,n}^{\mathrm{gr},(r,s)}\right)_{(\epsilon_{R})}$,
$\mathscr{PD}^{\mathscr{L}}$, $\mathscr{PD}^{\mathscr{R}}$,
$\mathscr{PD}^{{\mathscr{L}_{\bullet}}}$,
$\mathscr{PD}^{{\mathscr{R}_{\bullet}}}$,
$\tilde{\mathsf{L}}_{i,j}^{\mathrm{gr}}$.

Subsection 7.1: $\Gamma_{n}$, $\Gamma_{k,l,n}$.

Subsection 8.2: $u\star_{n}v$, $u\star_{k,l,n}v$.

## References

  * [BKV06] O. Bershtein, Y. Kolisnyk, L. Vaksman, _On a $q$-analogue of the Wallach-Okounkov formula._ Lett. Math. Phys. 78 (2006), no. 1, 97–109.
  * [CP94] V. Chari, A. N. Pressley, _A Guide to Quantum Groups_ , Cambridge University Press, Cambridge, 1994.
  * [EGNO15] P. Etingof, Pavel. S. Gelaki, D. Nikshych, V. Ostrik, _Tensor categories_ , Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.
  * [Ho95] R. Howe, _Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond._ The Schur lectures (1992) (Tel Aviv), 1–182, Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995.
  * [Ja96] J. C. Jantzen, _Lectures on quantum groups_. Graduate Studies in Mathematics, 6. American Mathematical Society, Providence, RI, 1996. viii+266.
  * [JL94] A. Joseph, G. Letzter, _Separation of variables for quantized enveloping algebras._ Amer. J. Math. 116 (1994), no. 1, 127–177.
  * [KS97] A. Klimyk, K. Schmudgen, _Quantum groups and their representations._ Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997. xx+552
  * [Kn97] F. Knop, _Symmetric and non-symmetric quantum Capelli polynomials._ Comment. Math. Helv. 72 (1997), no. 1, 84–100.
  * [LZZ11] G. I. Lehrer, H. Zhang, R. B. Zhang, _A quantum analogue of the first fundamental theorem of classical invariant theory_ , Commun. Math. Phys. 301 (2011), 131–174.
  * [LSS22a] G. Letzter, S. Sahi, H. Salmasian, _Weyl algebras and Capelli operators for quantum homogneous spaces_ , preprint.
  * [LSS22b] G. Letzter, S. Sahi, H. Salmasian, _The Capelli eigenvalue problem for quantum groups_ , preprint.
  * [NYM93] M. Noumi, H. Yamada, K. Mimachi, _Finite dimensional representations of the quantum group $GL_{q}(n,\mathbb{C})$ and the Zonal spherical functions on $U_{q}(n-1)\backslash U_{q}(n)$_, Japan. J. Math. (N.S.) 19 (1993), no. 1, 31–80.
  * [No96] M. Noumi, _Macdonald’s Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces_ , Adv. Math. 123 (1996), 16–77.
  * [Ok97] A. Okounkov, _Binomial formula for Macdonald polynomials and applications._ Math. Res. Lett. 4 (1997), no. 4, 533–553.
  * [RT10] A. Ram, P. Tingley, _Universal Verma modules and the Misra-Miwa Fock space_ , Int. Jour. Math. Math. Sci. 2010, Article ID 326247, 19 pp., doi:10.1155/2010/326247
  * [Sah96] S. Sahi, _Interpolation, integrality, and a generalization of Macdonald’s polynomials._ Internat. Math. Res. Notices 1996 (1996), no. 10, 457–471.
  * [Sah11] S. Sahi, _Binomial coefficients and Littlewood–Richardson coefficients for interpolation polynomials and Macdonald polynomials._ Representation theory and mathematical physics, 359–369, Contemp. Math., 557, Amer. Math. Soc., Providence, RI, 2011.
  * [SSV04] D. Shklyarov, S. Sinel’shchikov, L. Vaksman, _Fock representations and quantum matrices._ Internat. J. Math. 15 (2004), no. 9, 855–894.
  * [Ta92] M. Takeuchi, _Some topics on $GL_{q}(n)$._ J. Algebra 147 (1992), no. 2, 379–410.
  * [VV94] A. Van Daele, S. Van Keer, _The Yang-Baxter and pentagon equation_. Compositio Math. 91 (1994), no. 2, 201–221.
  * [VY20] C. Voigt, R. Yuncken, _Complex semisimple quantum groups and representation theory._ Lecture Notes in Mathematics, 2264. Springer, Cham, (2020). x+374 pp.
  * [Wy39] H. Weyl, _The Classical Groups. Their Invariants and Representations._ Princeton University Press, Princeton, N.J., 1939. xii+302 pp.
  * [Zh02] R. B. Zhang, _Howe duality and the quantum general linear group._ Proc. Amer. Math. Soc. 131 (2003), no. 9, 2681–2692.