A complete characterization of graphs for which $m_G(-1) = n-d-1$

TL;DR

This paper fully characterizes connected graphs where the multiplicity of the eigenvalue -1 equals the number of vertices minus the diameter minus one, including cases where -1 is an eigenvalue of the diameter path.

Contribution

It provides a complete characterization of graphs achieving the extremal eigenvalue multiplicity condition, including cases where -1 is an eigenvalue of the diameter path.

Findings

Graphs with $m_G(-1) = n - d - 1$ are fully characterized.

Extremal graphs are identified when $-1$ is or isn't an eigenvalue of the diameter path.

The results unify the understanding of eigenvalue multiplicities related to graph diameter.

Abstract

Let $G$ be a simple connected graph of order $n$ with diameter $d$. Let $m_G(-1)$ denote the multiplicity of the eigenvalue $-1$ of the adjacency matrix of $G$, and let $P = P_{d+1}$ be the diameter path of $G$. If $-1$ is not an eigenvalue of $P$, then by the interlacing theorem, we have $m_G(-1)\leq n - d - 1$. In this article, we characterize the extremal graphs where equality holds. Moreover, for the completeness of the results, we also characterize the graphs $G$ that achieve $m_G(-1) = n - d - 1$ when $-1$ is an eigenvalue of $P$. Thus, we provide a complete characterization of the graphs $G$ for which $m_G(-1) = n - d - 1$.