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

Qian-Qian Chen, Ji-Ming Guo

Abstract

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 $m_λ(G)=β'(G)+c(G)-1$.

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

Abstract

Let be a graph with an adjacent matrix . The multiplicity of an arbitrary eigenvalue of is denoted by . In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of at least , then for any . Moreover, they characterized all trees with , where is the induced matching number of . 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 of the connected graph , , where is the cyclomatic number of , and the equality holds if and only if or , or a tree with the diameter is at most . Furthermore, if , we characterize all connected graphs with .
Paper Structure (3 sections, 16 theorems, 28 equations, 2 figures)

This paper contains 3 sections, 16 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Result

Key Result

Theorem 1.1

Wong Let $T$ be a tree with $\lambda \neq 0$ as an eigenvalue of multiplicity $k \geq 1$. If the diameter of $T$ is at least $4$, then $T$ has $k+1$ pendant edges forming an induced matching of $T$. Particularly, $m_\lambda(T) \leq \beta^{\prime}(T)$, and $m_\lambda(T) \leq \beta^{\prime}(T)-1$ if t

Figures (2)

  • Figure 1: All possible forms of graph $C^*$.
  • Figure 2: A graph with $-2$ as an eigenvalue of multiplicity $\beta'(G)+c(G)-1$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 15 more