arXiv subjects

Songnian Xu

Publications and source records attributed to Songnian Xu.

15 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↗

arXiv2026

On automorphism group of the reduced finitary power monoid of the additive group of integers

Let $\mathbb{Z}$ be the additive group of all integers and $\mathbb{N}$ the sub-monoid of $\mathbb{Z}$ of all non-negative integers. For a finite subset $X$ of $\mathbb{Z}$, we denote by ${\rm max}\ X$ the maximum member in $X$. %Recently, Tringali and Yan (\cite{tri2}, J. Combin. Theory Ser. A, 209(2025)) proved that the only non-trivial automorphism of $\mathcal{P}_{{\rm fin,} 0}(\mathbb{N})$ %is the involution $X \mapsto \beta(X) - X$, and they posed a conjecture: {\it The automorphism group of the reduced power monoid $\mathcal{P}_{{\rm fin,} 0}(S)$ of a numerical %monoid $S$ properly contained in $\mathbb{N}$ must be the identity}. Recently, Tringali and Yan (\cite{tri2}, J. Comb. Theory, Ser. A, 209(2025)) proved that the only non-trivial automorphism of $\mathcal{P}_{{\rm fin,} 0}(\mathbb{N})$ is the involution $X \mapsto {\rm max}\ X - X$. Following up on the result in \cite{tri2}, Tringali and Wen \cite{triwen} proved that the automorphism group of the power monoid $\mathcal{P}_{\rm fin}(\mathbb{Z})$ is isomorphic to $\mathbb{Z}_2 \times {\rm Dih}_{\infty}$, where ${\rm Dih}_{\infty}$ refers to the infinite dihedral group. At the end part of \cite{triwen}, Tringali and Wen left a conjecture as follows: {\it The only non-trivial automorphism of the reduced finitary power monoid of $(\mathbb{Z},+)$ is given by $X\mapsto -X$.} In the present paper, we aim to give a positive proof for the above conjecture.

math.GR↗

arXiv2025

On automorphism groups of power semigroups over numerical semigroups or over numerical monoids

A numerical semigroup $S$ is a cofinite subsemigroup of $ \mathbb{N}$, where $\mathbb{N}$ is the additive monoid of non-negative integers. Denote by $\mathcal{P}_{\rm fin} (S)$ the semigroup consisting of all non-empty finite subsets of $S$ endowed with the operation of setwise addition defined by $$X+Y=\{x+y:x\in X, y\in Y\}, \qquad\text{for all } X, Y \in \mathcal P_\text{fin}(S).$$ We call $\mathcal{P}_{\rm fin} (S)$ the finitary power semigroup of $S$. When $0\in S$ (and hence $S$ is a numerical monoid), the family $\mathcal P_{\text{fin},0}(S)$ of all finite subsets of $S$ containing $0$ is a submonoind of $\mathcal P_\text{fin}(S)$; we call $\mathcal{P}_{{\rm fin}, 0}(S)$ the reduced finitary power monoid of $S$ with the singleton $\{0\}$ as zero-element. For a non-empty finite subset $X$ of $\mathbb{N}$, we denote by $ \min X$ and $\max X $ the minimum and the maximum in $X$. Tringali and Yan have recently proved in [J.\ Combin.\ Theory Ser.\ A 209 (2025)] that the only non-trivial automorphism of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$ is the involution $X \mapsto \max X - X$. By applying Tringali-Yan's result, we in this article determined the automorphism group of the finitary power semigroup $\mathcal{P}_{\rm fin}(S)$ of an arbitrary numerical semigroup $S$. More precisely, if $S$ is the set of all integers larger than or equal to a fixed $k \in \mathbb N$, then the only non-trivial automorphism of $\mathcal{P}_{\rm fin}(S)$ is the involution $X \mapsto \max X - X+ \min X$; otherwise, $\mathcal{P}_{\rm fin}(S)$ has only the identity automorphism.

math.GR↗

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 $\Gamma$ over $G$ such that $\Gamma$ is oriented and $\mathrm{Aut}(\Gamma) \cong G$. 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}) $\Gamma$ with $\mathrm{Aut}(\Gamma) \cong G$. 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

On oriented $m$-semiregular representations of finite groups about valency three

Let $G$ be a group and $m$ a positive integer. We say an $m$-Cayley digraph $\Gamma$ over $G$ is a digraph that admits a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. The digraph $\Sigma$ is $k$-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 that $G$ admits an oriented $m$-semiregular representation (abbreviated as OmSR) if there exists a regular $m$-Cayley digraph $\Gamma$ over $G$ such that $\Gamma$ is oriented and its automorphism group is isomorphic to $G$. In particular, an O1SR is called an ORR. Xia et al. \cite{x2} provided a classification of finite simple groups admitting an ORR of valency 2. Furthermore, in 2022, Du et al. \cite{du2} proved that most finite simple groups admit an OmSR of valency 2 for $m \geq 2$, except for a few exceptional cases. In this paper, we classify the finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$.

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↗

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 $m$-$Cayley$ $digraph$ $\Gamma = \mathrm{Cay}(G, T_{i,j}: i,j \in \mathbb{Z}_m)$ 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 $\Gamma$ 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 $\Sigma$ 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 $\Gamma$ with $\mathrm{Aut}(\Gamma) \cong G$. 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

Finite groups admitting a regular tournament $m$-semiregular representation

For a positive integer $m$, a finite group $G$ is said to admit a tournament $m$-semiregular representation (TmSR for short) if there exists a tournament $\Gamma$ such that the automorphism group of $\Gamma$ is isomorphic to $G$ and acts semiregularly on the vertex set of $\Gamma$ with $m$ orbits. Clearly, every finite group of even order does not admit a TmSR for any positive integer $m$, and T1SR is the well-known tournament regular representation (TRR for short). In 1986, Godsil \cite{god} proved, by a probabilistic approach, that the only finite groups of odd order without a TRR are $\mathbb{Z}_3^2$ and $\mathbb{Z}_3^3$ . More recently, Du \cite{du} proved that every finite group of odd order has a TmSR for every $m \geq 2$. The author of \cite{du} observed that a finite group of odd order has no regular TmSR when $m$ is an even integer, a group of order $1$ has no regular T3SR, and $\mathbb{Z}_3^2$ admits a regular T3SR. At the end of \cite{du}, Du proposed the following problem. \noindent{\sf\it Problem.} \ \ {\it For every odd integer $m\geq 3$, classify finite groups of odd order which have a regular TmSR.} The motivation of this paper is to give an answer for the above problem. We proved that if $G$ is a finite group with odd order $n>1$, then $G$ admits a regular TmSR for any odd integer $m\geq 3$.

math.GR↗

arXiv2024

Line graphs with the largest eigenvalue multiplicity

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

math.SP↗

arXiv2024

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

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 $\lambda$ is not an eigenvalue of $P_s$ and $G$ is neither a cycle nor a starlike tree $T_k$, it holds that $m_G(\lambda) \leq 2c(G) + q_s(G) - 1$ and characterized the extremal graphs when $G$ is a tree. In this article, we characterize the extremal graphs for which $m_G(\lambda) = 2c(G) + q_s(G) - 1$ when $G \in \mathbb{G}_{s}$ and $\lambda\notin \sigma(P_s)$.

math.SP↗

arXiv2024

A characterization of graphs $G$ with nullity $n(G)-d(G)-1$

For a connected graph $G$ with order $n$, let $e(G)$ represent the number of its distinct eigenvalues, and let $d$ denote its diameter. We denote the eigenvalue multiplicity of $\mu$ in $G$ by $m_G(\mu)$. It is well established that the inequality $e(G) \geq d + 1$ implies that when $\mu$ is an eigenvalue of $P_{d+1}$, it follows that $m_G(\mu) \leq n - d$; otherwise, for any real number $\mu$, we have $m_G(\mu) \leq n - d - 1$. A graph is termed minimal if $e(G) = d + 1$. In 2013, Wong et al. characterized all minimal graphs for which $m_G(0) = n - d$. In this article, we provide a complete characterization of the graphs $G$ such that $m_G(0) = n - d - 1$.

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

Near automorphisms of $G_{(n,m)}$

Let $G$ be a graph with vertex set $V(G)$, $f$ a permutation of $V(G)$. Define $δ_f(G)=|d(x,y)-d(f(x),f(y))|$ and $δ_f(G)=Σδ_f(x,y)$, where the sum is taken over all unordered pair $x$, $y$ of distinct vertices of $G$. $δ_f(x,U)=Σδ_f(x,y)$, where $U\subseteq V(G)$ and $y\in U$. Let $π(G)$ denote the smallest positive value of $δ_f(G)$ among all permutations of $V(G)$. A permutation $f$ with $δ_f(G)=π(G)$ is called a near automorphisms of $G$\cite{HV}. In this paper, we define $G_{(n,m)}$ is a graph obtained from $K_n$ by add $t_i$ pendent vertices to $y_i$ which is a vertex of $K_n$, $i=1,\cdots,m$, and we say $y_i$ is a c-pendent vertex of $G_(n,m)$. We determine $π(G_{(n,m)})$ and describe permutations $f$ of $G_{(n,m)}$ for which $π(G_{(n,m)})=δ_f(G_{(n,m)})$. Because $G_{(1,1)}$ is a star and it is easy, hence we let $n\geq 2$. Suppose $G_(n,m)$ has $m$ c-pendent vertices $\{y_1, \ldots, y_m\}$ and $y_i$ has $t_i$ pendent vertices($1\leq t_1\leq t_2\leq \ldots \leq t_m$). For $m<n$ we have $$π(G_{(n,m)})= \left\{ \begin{array}{lc} 2n-4 & n \leq t_1+2, m=1 \cr 2t_1&otherwise \end{array} \right. $$ For $m=n$ we have $$π(G_{(n,n)})= \left\{ \begin{array}{lc} 4 & t_1=1,t_2=2 \cr 2t_1+2t_2&otherwise \end{array} \right.$$

math.CO↗