A characterization of graphs with two types of eigenvalue multiplicities equal to n−d−1

Abstract

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