The multiplicity of the laplacian eigenvalue 1 of a tree

Abstract

Let $G$ be a connected, undirected simple graph. Denote by $L(G)$ the Laplacian matrix of $G$, and let $m_G(\lambda )$ be the multiplicity of an eigenvalue $\lambda$ of $L(G)$. When $G$ is a tree $T$ with $n\ge 6$ vertices, Tian et al. [Discrete Mathematics, 2026] proved that if $T$ is reduced and contains no pendant $P_3$, then \[ m_{T}(1) \le \frac{n-6}{4}, \] and they gave a complete characterization of the graphs for which equality holds. In this paper, we further investigate the above problem. Still assuming that $T$ is a tree with $n\ge 7$ vertices which is reduced and has no pendant $P_3$, we prove the following results. If $m_T(1)\ne \frac{n-6}{4}$, then \[ m_{T}(1) \le \frac{n-7}{4}, \] and we give a complete characterization of the graphs for which equality holds. If, moreover, $m_T(1)\ne \frac{n-6}{4},\frac{n-7}{4}$, then \[ m_{T}(1) \le \frac{n-8}{4}, \] and we also give a complete characterization of the extremal graphs.