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

TL;DR

This paper characterizes graphs where the eigenvalue zero has multiplicity exactly one less than the maximum possible, extending previous classifications of minimal graphs with specific eigenvalue multiplicities.

Contribution

It provides a complete characterization of graphs with eigenvalue zero multiplicity equal to n - d - 1, a case not previously fully described.

Findings

Characterization of graphs with m_G(0) = n - d - 1

Extension of previous minimal graph classifications

Insight into eigenvalue multiplicity bounds in graphs

Abstract

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 $m_G(0) = n - d - 1$.