arXiv subjects

Ji-Ming Guo

Publications and source records attributed to Ji-Ming Guo.

6 recordsLinked to original sources

arXiv2026

Extremal results on the second largest eigenvalue of graphs with given order

In this paper, we demonstrate the effects on the second largest eigenvalue $\lambda_2(G)$ of a connected graph $G$ after edge addition or deletion. In 1989, Chung, Graham and Wilson showed $\max\{|\lambda_2|,|\lambda_n|\}>\Omega(n)$ for dense $K_{r+1}$-free graphs of order $n$, giving spectral comprehension of existence of large clique or independent set, respect to Ramsey theory. Applying the results of effects on $\lambda_2$ after edge operations, we determine the maximum value of $\lambda_2$ among all $K_{r+1}$-free connected graphs with given order, and completely characterize the extremal graphs. Moreover, for arbitrary given graph $F$, we investigates the maximum second largest $\lambda_2(G)$ among $F$-free connected graphs of order $n$. Let $\rho^*(n,F)$ be the maximum spectral radius of $F$-free graphs on $n\ge n_F$ vertices, and $G^*(n,F)$ be a graph with its spectral radius $\rho\big(G^*(n,F)\big)=\rho^*(n,F)$. We prove that, for an $F$-free connected graph $G$ of order $n\ge f(n_F)$, \\(1) if $n$ is odd, then $$\lambda_2(G)\le\rho^*\left(\frac{n-1}{2},F\right)$$ with equality if and only if $G\in \mathcal{I}\big(G^*(\frac{n-1}{2},F),G^*(\frac{n-1}{2},F)\big)$; and\\ (2) if $n$ is even, and $F$ does not contain cut edges, then the graph $G^\dag$ with the maximum second largest eigenvalue satisfies $$\lambda_2(G^\dag)=\rho^*\left(\frac{n}{2},F\right)-o(1)$$ and $G^\dag\in \mathcal{E}\big(H_1,H_2\big)$, where $H_1$ and $H_2$ are $F$-saturated graphs on $\frac{n}{2}$ vertices. In particular, other than a complete graph $K_{r+1}$, when $F$ is a book graph $B_{k+1}$ or an odd cycle $C_{2k+1}$, we are able to determine the maximum second largest eigenvalue for $F$-free connected graphs of given order, and completely characterize the extremal graphs.

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arXiv2026

Upper bounds of the second largest eigenvalue of graphs

Let $\lambda_i(G)$ denote the $i$-th largest eigenvalue of adjacency matrix of a graph $G$. Gerschgorin's Theorem indicates $\lambda_1(G)$ belongs to the largest disk, i.e., $\lambda_1(G)\le\Delta_1(G)$, where $\Delta_i(G)$ is the $i$-th largest degree of $G$. We show that $\lambda_2(G)$ lies in the second largest disk. That is, in detail, $$\lambda_2(G)<\Delta_2(G)-\frac{1}{n^2}.$$ A classical theorem proved by Hong [\textit{Linear Algebra Appl.} 1988] states that $\lambda_1(G)\le\sqrt{2m-n+1}$ for a connected graph $G$ with $n$ vertices and $m$ edges, where the equality holds if and only if $G$ is a star $S_n$ or a complete graph $K_n$. We give a refinement of Hong's theorem by showing $$\lambda_1(G)<\sqrt{2m-n}$$ for any connected graph $G\not\in\left\{S_n,S^1_{n-1},K_n,K^1_{n-1}\right\}$. Based on this improved upper bound of $\lambda_1(G)$, for a connected graph $G$ with $n$ vertices and $m$ edges, we are able to prove a sharp upper bound of $\lambda_2(G)$ that $$\lambda_2(G)\le\sqrt{m-\frac{n}{2}-\frac{1}{2}},$$ except $G$ is obtained from two disjoint $S_\frac{n}{2}$ by adding an edge between a pendant vertex of each star. Moreover, we provide a complete characterization to extremal graphs attaining the equality.

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arXiv2024

A relation between multiplicity of nonzero eigenvalues and the matching number of graph

Let $G$ be a graph with an adjacent matrix $A(G)$. The multiplicity of an arbitrary eigenvalue $λ$ of $A(G)$ is denoted by $m_λ(G)$. In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of $T$ at least $4$, then $m_λ(T)\leq β'(T)-1$ for any $λ\neq0$. Moreover, they characterized all trees with $m_λ(T)=β'(T)-1$, where $β'(G)$ is the induced matching number of $G$. In this paper, we intend to extend this result from trees to any connected graph. Contrary to the technique used in \cite{Wong}, we prove the following result mainly by employing algebraic methods: For any non-zero eigenvalue $λ$ of the connected graph $G$, $m_λ(G)\leq β'(G)+c(G)$, where $c(G)$ is the cyclomatic number of $G$, and the equality holds if and only if $G\cong C_3(a,a,a)$ or $G\cong C_5$, or a tree with the diameter is at most $3$. Furthermore, if $β'(G)\geq3$, we characterize all connected graphs with $m_λ(G)=β'(G)+c(G)-1$.

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arXiv2023

The multiplicity of a Hermitian eigenvalue on graphs

For a graph $G$, let $\mathcal{S}(G)$ be the set consisting of Hermitian matrices whose graph is $G$. Denoted by $m_B(G,λ)$ the multiplicity of an eigenvalue $λ$ of $B(G)\in \mathcal{S}(G)$, we show that $m_B(G,λ)\le 2θ(G)+p(G)$ where $θ(G)$ and $p(G)$ are the cyclomatic number and the number of pendent vertices of $G$ respectively, and characterize the graphs attaining the equality. This is a generalization of a result on adjacency matrix by Wang et al.\cite{Wang1}. Moreover, they arose an open problem in \cite{Wang1}: \textit{characterize all graphs with $m_A(G,λ)=2θ(G)+p(G)-1$ for any eigenvalue $λ$ of its adjacency matrix.} In this paper, we completely characterize the graphs with $m_B(G,λ)=2θ(G)+p(G)-1$ for any eigenvalue $λ$ of an arbitrary Hermitian matrix $B(G)\in \mathcal{S}(G)$. This result provides a stronger answer to the above problem, and encompasses some previous known works considering $λ=-1$ or $0$ on the problem.

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arXiv2023

The spectral radius of graphs with fractional matching number

Let $\mathcal{G}_{n, β^*}$ $(\mathcal{G}^*_{n,β^*})$ be the set of all (connected) graphs of order $n$ with fractional matching number $β^*$. In this paper, the graphs with maximal spectral radius in $\mathcal{G}_{n,β^*}$ and $\mathcal{G}^*_{n,β^*}$ are characterized, respectively. Moreover, a lower bound for the spectral radius in graphs with order $n$ to guarantee the existence of a perfect fractional matching is also given, which generalizes the main result of O [Suil O, Spectral radius and matchings in graphs, Linear Algebra and its Applications, 2020].

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arXiv2022

Maximum degree and spectral radius of graphs in terms of size

Research on the relationship of the (signless Laplacian) spectral radius of a graph with its structure properties is an important research project in spectral graph theory. Denote by $ρ(G)$ and $q(G)$ the spectral radius and the signless Laplacian spectral radius of a graph $G$, respectively. Let $k\ge 0$ be a fixed integer and $G$ be a graph of size $m$ which is large enough. We show that if $ρ(G)\ge\sqrt{m-k}$, then $C_4\subseteq G$ or $K_{1,m-k}\subseteq G$. Furthermore, we prove that if $q(G)\ge m-k$, then $K_{1,m-k}\subseteq G$. Both these two results extend some known results.

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