A characterization of graphs $G$ with $m_G(\lambda)= 2c(G) + q_s(G) - 1$

TL;DR

This paper characterizes graphs with a specific eigenvalue multiplicity relation involving cycle count and pendant vertex sets, extending previous results to a broader class of graphs.

Contribution

It provides a complete characterization of extremal graphs satisfying the eigenvalue multiplicity condition within the class G_s, generalizing prior work on trees.

Findings

Identifies extremal graphs where eigenvalue multiplicity equals 2c(G) + q_s(G) - 1.

Extends characterization from trees to graphs with pendant paths of length at least P_s.

Clarifies spectral properties for graphs with pendant path constraints.

Abstract

Let $G$ be a simple connected graph. If every pendant path in $G$ is at least $P_s$, we denote that $G\in \mathbb{G}_s$. For $G \in \mathbb{G}_s$, let $Q_s(G)$ be the set of vertices in $G$ that are distance $s$ from the pendant vertex, and let $|Q_s(G)| = q_s(G)$. For $G \in \mathbb{G}_s$, Li et al. (2024) proved that when $\lambda$ is not an eigenvalue of $P_s$ and $G$ is neither a cycle nor a starlike tree $T_k$, it holds that $m_G(\lambda) \leq 2c(G) + q_s(G) - 1$ and characterized the extremal graphs when $G$ is a tree. In this article, we characterize the extremal graphs for which $m_G(\lambda) = 2c(G) + q_s(G) - 1$ when $G \in \mathbb{G}_{s}$ and $\lambda\notin \sigma(P_s)$.