Multiplicity of Laplacian eigenvalue 1 of a graph

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 $G$. A graph $G$ is called reduced, if $p(G)=q(G)$. It is known that deleting a pendant path $P_3$ from a graph $G$ cannot change $m_{L(G)}(1)$. By the reduction operation for a graph (defined by Tian and Wong, 2026), we could turn to the reduced graphs with each quasi-pendant vertex of degree 2 to investigate $m_{L(G)}(1)$. Then let $T$ be a reduced tree on $n(\ge 7)$ vertices with each quasi-pendant vertex of degree 2 and without pendant path $P_3$. We first prove that \begin{equation*} m_{L(T)}(1)\leq \frac{n-5}{6} \end{equation*} and the extremal trees attaining the upper bound are determined completely. In addition, let $G$ be an arbitrary connected reduced graph with order $n\ge 6$ and size $m$. Denote by $c=m-n+1$ the first Betti number of $G$, then we obtain \begin{equation*} m_{L(G)}(1)\leq c+\frac{n-2}{4}, \end{equation*} and the extremal graphs attaining the upper bound are characterized completely.