arXiv subjects

Wenhao Zhen

Publications and source records attributed to Wenhao Zhen.

10 recordsLinked to original sources

arXiv2026

Graphs with maximal Laplacian eigenvalue multiplicity

In this paper, $ G $ is a simple connected graph, and $ m_G(\lambda) $ denotes the multiplicity of $ \lambda $ as an eigenvalue of the Laplacian matrix $ L(G) $. Let $ p(G) $ denote the number of pendant vertices of $ G $, $ q(G) $ the number of quasi-pendant vertices of $ G $, and $ c(G) $ the dimension of the cycle space of $ G $. Li et al. [Discrete Mathematics, 2026] proved that if $ G $ is a tree with $ G \not\cong K_{1,n-1} $ and $ \lambda \neq 1 $, then \[ m_G(\lambda) \le q(T) - 1. \] Moreover, for a general graph $ G $ with $ \lambda \neq 1 $, Li et al. also proved in the same paper that \[ m_G(\lambda) \le c(G) + q(G), \] with equality if and only if $ G \cong K_{1,n-1} $ or $ G \cong C_n $ with $ \lambda \notin \{0, 4\} $. A natural consequence is that if $ c(G) + q(G) \ge 2 $, then \[ m_G(\lambda) \le 2c(G) + q(G) - 1. \] When $ c(G) = 0 $, this reduces to the result of Li et al. for trees. In this paper, we give a complete characterization of graphs $ G $ attaining the equality \[ m_G(\lambda) = 2c(G) + q(G) - 1. \]

math.SP↗

arXiv2026

The multiplicity of the laplacian eigenvalue 1 of a tree

Let $G$ be a connected, undirected simple graph. Denote by $L(G)$ the Laplacian matrix of $G$, and let $m_{G}(\lambda)$ be the multiplicity of an eigenvalue $\lambda$ of $L(G)$. When $G$ is a tree $T$ with $n \ge 6$ vertices, Tian et al. [Discrete Mathematics, 2026] proved that if $T$ is reduced and contains no pendant $P_3$, then \[ m_{T}(1) \le \frac{n-6}{4}, \] and they gave a complete characterization of the graphs for which equality holds. In this paper, we further investigate the above problem. Still assuming that $T$ is a tree with $n \ge 7$ vertices which is reduced and has no pendant $P_3$, we prove the following results. If $m_T(1) \neq \frac{n-6}{4}$, then \[ m_{T}(1) \le \frac{n-7}{4}, \] and we give a complete characterization of the graphs for which equality holds. If, moreover, $m_T(1) \neq \frac{n-6}{4}, \frac{n-7}{4}$, then \[ m_{T}(1) \le \frac{n-8}{4}, \] and we also give a complete characterization of the extremal graphs.

math.SP↗

arXiv2025

$m$-partite oriented semiregular representation of valency 3 for finite groups

Let $G$ be a finite group and $m \geq 2$ a positive integer. We say that $G$ admits an \emph{oriented $m$-semiregular representation} (abbreviated as OmSR) if there exists a $m$-Cayley digraph $Γ$ over $G$ such that $Γ$ is oriented and $\mathrm{Aut}(Γ) \cong G$. In \cite{xu1}, we classified finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$ and $G \ncong \mathbb{Z}_1$. In this article, we consider $m$-partite digraphs.We say a finite group $G$ admits an \emph{$m$-partite oriented semiregular representation} ($m$-partite digraphical representation), abbreviated as \emph{$m$-POSR} (\emph{$m$-PDR}), if there exists an \emph{oriented} $m$-partite Cayley digraph (\emph{$m$-partite Cayley digraph}) $Γ$ with $\mathrm{Aut}(Γ) \cong G$. In this paper, we classify finite groups generated by at most two elements that admit $m$-POSR. Since if $G$ admits an $m$-POSR, then $G$ must also admit an $m$-PDR (while the converse does not hold), as a natural consequence, we also provide a complete classification for groups $G=\langle x,y\rangle$ that admit $m$-PDR of valency 3. This complements the results in \cite{xu2}.

math.GR↗

arXiv2025

The $m$-partite digraphical representations of valency 3 of finite groups generated by two elements

Let $G$ be a finite group and $m$ be an integer. We employ the notation $g_i$ to represent elements $(g,i)$ in the Cartesian product $G \times \mathbb{Z}_m$, where $\mathbb{Z}_m$ denotes integers modulo $m$. For given sets $T_{i,j} \subseteq G$ ($i,j \in \mathbb{Z}_m$), we construct the $m$-$Cayley$ $digraph$ $Γ= \mathrm{Cay}(G, T_{i,j}: i,j \in \mathbb{Z}_m)$ with vertex set $\bigcup_{i\in\mathbb{Z}_m}G_i$ (where $G_i = \{g_i | g \in G\}$) and arc set $\bigcup_{i,j}\{(g_i, (tg)_j) | t \in T_{i,j}, g \in G\}$. When $T_{i,i} = \emptyset$ for all $i \in \mathbb{Z}_m$, we call $Γ$ an \emph{$m$-partite Cayley digraph}. For $m$-partite Cayley digraphs, we observe that a $1$-partite Cayley digraph is necessarily an empty graph. Therefore, throughout this paper, we restrict our consideration to the case where $m \geq 2$. The digraph $Σ$ is regular if there exists a non-negative integer $k$ such that every vertex has out-valency and in-valency equal to $k$. All digraphs considered in this paper are regular. We say a group $G$ admits an \emph{$m$-partite digraphical representation} ($m$-PDR for short) if there exists a regular $m$-partite Cayley digraph $Γ$ with $\mathrm{Aut}(Γ) \cong G$. Based on Du et al.'s complete classification of unrestricted $m$-PDRs \cite{du4} (2022), we focus on the unresolved valency-specific cases. In this paper, we investigate $m$-PDRs of valency 3 for groups generated by at most two elements, and establish a complete classification of nontrivial finite simple groups admitting $m$-PDRs of valency 3 with $m\geq2$.

math.GR↗

arXiv2025

Laplacian eigenvalue distribution and girth of graphs

Let $G$ be a connected graph on $n$ vertices with girth $g$. Let $m_GI$ denote the number of Laplacian eigenvalues of graph $G$ in an interval $I$. In this paper, we show that if $G$ is not a cycle, then $m_G(n-g+3,n]\leq n-g$. Moreover, we prove that $m_G(n-g+3,n]= n-g$ if and only if $G\cong C_3$ or $G\cong K_{3,2}$ or $G\cong U_1$, where $U_1$ is obtained from a cycle by joining a single vertex with a vertex of this cycle.

math.CO↗

arXiv2024

Line graphs with the largest eigenvalue multiplicity

For a connected graph $G$, we denote by $L(G)$, $m_{G}(λ)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $λ$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In 2023, Yang et al. \cite{WL LT} proved that $m_{L(T)}(λ)\leq p(T)-1$ for any tree $T$ with $p(T)\geq 3$, and characterized all trees $T$ with $m_{L(T)}(λ) = p(T)-1$. In 2024, Chang et al. \cite{-1 LG} proved that, if $G$ is not a cycle, then $m_{L(G)}(λ)\leq 2c(G)+p(G)-1$, and characterized all graphs $G$ with $m_{L(G)}(-1) = 2c(G)+p(G)-1$. The remaining ploblem is to characterize all graphs $G$ with $m_{L(G)}(λ)= 2c(G)+p(G)-1$ for an arbitrary eigenvalue $λ$ of $L(G)$. In this paper, we give this problem a complete solution.

math.SP↗

arXiv2024

A characterization of graphs $G$ with $m_G(λ)= 2c(G) + q_s(G) - 1$

Let $G$ be a simple connected graph. If every pendant path in $G$ is at least $P_s$, we denote that $G\in \mathbb{G}_s$. For $G \in \mathbb{G}_s$, let $Q_s(G)$ be the set of vertices in $G$ that are distance $s$ from the pendant vertex, and let $|Q_s(G)| = q_s(G)$. For $G \in \mathbb{G}_s$, Li et al. (2024) proved that when $λ$ is not an eigenvalue of $P_s$ and $G$ is neither a cycle nor a starlike tree $T_k$, it holds that $m_G(λ) \leq 2c(G) + q_s(G) - 1$ and characterized the extremal graphs when $G$ is a tree. In this article, we characterize the extremal graphs for which $m_G(λ) = 2c(G) + q_s(G) - 1$ when $G \in \mathbb{G}_{s}$ and $λ\notin σ(P_s)$.

math.SP↗

arXiv2024

A complete characterization of graphs for which $m_G(-1) = n-d-1$

Let $G$ be a simple connected graph of order $n$ with diameter $d$. Let $m_G(-1)$ denote the multiplicity of the eigenvalue $-1$ of the adjacency matrix of $G$, and let $P = P_{d+1}$ be the diameter path of $G$. If $-1$ is not an eigenvalue of $P$, then by the interlacing theorem, we have $m_G(-1)\leq n - d - 1$. In this article, we characterize the extremal graphs where equality holds. Moreover, for the completeness of the results, we also characterize the graphs $G$ that achieve $m_G(-1) = n - d - 1$ when $-1$ is an eigenvalue of $P$. Thus, we provide a complete characterization of the graphs $G$ for which $m_G(-1) = n - d - 1$.

math.SP↗

arXiv2024

The relationship between the negative inertia index of graph $G$ and its girth $g$ and diameter $d$

Let $G$ be a simple connected graph. We use $n(G)$, $p(G)$, and $η(G)$ to denote the number of negative eigenvalues, positive eigenvalues, and zero eigenvalues of the adjacency matrix $A(G)$ of $G$, respectively. In this paper, we prove that $2n(G)\geq d(G) + 1$ when $d(G)$ is odd, and $n(G) \geq \lceil \frac{g}{2}\rceil - 1$ for a graph containing cycles, where $d(G)$ and $g$ are the diameter and girth of the graph $G$, respectively. Furthermore, we characterize the extremal graphs for the cases of $2n(G) = d(G) + 1$, $n(G) = \lceil \frac{g}{2}\rceil$, and $n(G) = \lceil \frac{g}{2}\rceil - 1$.

math.SP↗

arXiv2023

Minimal graphs with eigenvalue multiplicity of $n-d$

For a connected graph $G$ with order $n$, let $e(G)$ be the number of its distinct eigenvalues and $d$ be the diameter. We denote by $m_G(μ)$ the eigenvalue multiplicity of $μ$ in $G$. It is well known that $e(G)\geq d+1$, which shows $m_G(μ)\leq n-d$ for any real number $μ$. A graph is called $minimal$ if $e(G)= d+1$. In 2013, Wang (\cite{WD}, Linear Algebra Appl.) characterize all minimal graphs with $m_G(0)=n-d$. In 2023, Du et al. (\cite{Du}, Linear Algebra Appl.) characterize all the trees for which there is a real symmetric matrix with nullity $n-d$ and $n-d-1$. In this paper, by applying the star complement theory, we prove that if $G$ is not a path and $m_G(μ)= n-d$, then $μ\in \{0,-1\}$. Furthermore, we completely characterize all minimal graphs with $m_G(-1)=n-d$.

math.SP↗