标题： 树$T$的特征与$m(T, λ)=p(T)-2$ 显示英文标题

标题： A characterization on trees $T$ with $m(T, λ)=p(T)-2$

摘要： 设$m(G,\lambda)$为连通图$G$的特征值$\lambda$的重数。 Wang 等 [Linear Algebra Appl. 584(2020), 257-266] 证明了对于任何连通图$G\neq C_n$，$m(G, \lambda) \leq 2c(G) + p(G) -1$，其中$c (G) = |E(G)| - |V (G)| + 1$和$p(G)$分别是$G$的环数和悬挂顶点的数量。 在同一篇论文中，他们提出了一个问题，即表征所有连通图$G$，其特征值为$\lambda$，使得$m(G, \lambda) =2c (G)+ p(G)-1$。Wong 等人。[离散数学。347(2024), 113845] 解决了当$G$是树的情况，通过表征所有树$T$，其特征值为$\lambda$，使得$m(T , \lambda) = p(T )-1$。 在本文中，我们进一步提供了关于树 $T$ 的结构表征，其特征值为 $\lambda$，使得 $m(T , \lambda) = p(T )-2$。 显示英文摘要

摘要： Let $m(G,\lambda)$ be the multiplicity of an eigenvalue $\lambda$ of a connected graph $G$. Wang et al. [Linear Algebra Appl. 584(2020), 257-266] proved that for any connected graph $G\neq C_n$, $m(G, \lambda) \leq 2c(G) + p(G) -1$, where $c (G) = |E(G)| - |V (G)| + 1$ and $p(G)$ are the cyclomatic number and the number of pendant vertices of $G$, respectively. In the same paper, they proposed the problem to characterize all connected graphs $G$ with eigenvalue $\lambda$ such that $m(G, \lambda) =2c (G)+ p(G)-1$. Wong et al. [Discrete Math. 347(2024), 113845] solved this problem for the case when $G$ is a tree by characterizing all trees $T$ with eigenvalue $\lambda$ such that $m(T , \lambda) = p(T )-1$. In this paper, we further provide the structural characterization on trees $T$ with eigenvalue $\lambda$ such that $m(T , \lambda) = p(T )-2$.