On the multiplicity of 1 as a Laplacian eigenvalue of a graph

TL;DR

This paper investigates the multiplicity of the eigenvalue 1 in the Laplacian spectrum of graphs, providing bounds and characterizations for trees and unicyclic graphs without pendant paths P3.

Contribution

It establishes a formula relating eigenvalue multiplicity to pendant vertices and reduces the problem to graphs without pendant path P3, with bounds and complete characterizations for specific graph classes.

Findings

Multiplicity of 1 in Laplacian spectrum is bounded by graph parameters.

Characterization of trees and unicyclic graphs attaining the bounds.

Reduction of the problem to graphs without pendant path P3.

Abstract

Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(\lambda)$ the multiplicity of $\lambda$ as a Laplacian eigenvalue of $G$. Let $\overline{G}$ be the reduced graph of $G$, which can be obtained from $G$ by deleting some pendant vertices such that $p(\overline{G})=q(\overline{G})$. We first prove that $m_{L(G)}(1)=p(G)-q(G)+m_{L(\overline{G})}(1)$. Since deleting pendant path $P_3$ does not change the multiplicity of Laplacian eigenvalue 1 of a graph, we further focus on reduced graphs without pendant path $P_3$. Let $T$ be a reduced tree on $n(\geq 6)$ vertices without pendant path $P_3$, then it is proved that $$m_{L(T)}(1)\leq \frac{n-6}{4},$$ and all the trees attaining the upper bound are characterized completely. As an application, for a reduced unicyclic graph $G$ of order $n\geq 10$ without pendant path $P_3$, we get $$m_{L(G)}(1)\leq \frac{n}{4},$$ and all the unicyclic graphs attaining the upper bound are determined completely.