The multiplicity of a Hermitian eigenvalue on graphs

TL;DR

This paper investigates the maximum multiplicity of a Hermitian eigenvalue in graphs, generalizing previous adjacency matrix results and characterizing graphs with near-maximum multiplicity for any eigenvalue.

Contribution

It extends known bounds on eigenvalue multiplicities to all Hermitian matrices associated with a graph and characterizes graphs achieving these bounds.

Findings

Established an upper bound on eigenvalue multiplicity involving cyclomatic number and pendent vertices.

Characterized graphs where the eigenvalue multiplicity reaches the bound minus one.

Provided a comprehensive solution to an open problem on eigenvalue multiplicities for all Hermitian matrices.

Abstract

For a graph $G$, let $\mathcal{S}(G)$ be the set consisting of Hermitian matrices whose graph is $G$. Denoted by $m_B(G,\lambda)$ the multiplicity of an eigenvalue $\lambda$ of $B(G)\in \mathcal{S}(G)$, we show that $m_B(G,\lambda)\le 2\theta(G)+p(G)$ where $\theta(G)$ 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 $m_A(G,\lambda)=2\theta(G)+p(G)-1$ for any eigenvalue $\lambda$ of its adjacency matrix.} In this paper, we completely characterize the graphs with $m_B(G,\lambda)=2\theta(G)+p(G)-1$ for any eigenvalue $\lambda$ 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 $\lambda=-1$ or $0$ on the problem.