ABSTRACT

Let T be a tree with vertex set $V\left(T\right)=\{v_1,v_2,\dots ,v_n\}$. The adjacency matrix $A\left(T\right)$ of T is an $n\times n$ matrix $\left(a_{ij}\right)$, where $a_{ij}=a_{ji}=1$ if $v_i$ is adjacent to $v_j$ and $a_{ij}=0$ if otherwise. In this paper, we consider the multiplicity of $-1$ as an eigenvalue of $A\left(T\right)$, which is written as $m(T,-1)$. It is proved that among all trees T with $p\ge 2$ pendant vertices, the maximum value of $m(T,-1)$ is p−1, and for a tree T with $p\ge 2$ pendant vertices, $m(T,-1)=p-1$ if and only if $T=P_n$ with $n\equiv 2\left(\mathrm{m}\mathrm{o}\mathrm{d}3\right)$, or T is a tree in which $d(v,u)\equiv 2\left(\mathrm{m}\mathrm{o}\mathrm{d}3\right)$ for any pendant vertex v and any major vertex u of T, where a major vertex is a vertex of degree at least 3 and $d(v,u)$ is the distance between v and u.

摘要

令T为一棵有顶点集的树$五\left(吨\right)=\{v_1,v_2,\dots ,v_n\}$. 邻接矩阵$\mathrm{一个}\left(吨\right)$T是一个$n\times n$矩阵$\left({\mathrm{一个}}_{\mathrm{一世}j}\right)$， 在哪里${\mathrm{一个}}_{\mathrm{一世}j}={\mathrm{一个}}_{j\mathrm{一世}}=1$如果$v_{\mathrm{一世}}$毗邻$v_j$和${\mathrm{一个}}_{\mathrm{一世}j}=0$否则。在本文中，我们考虑多重性$-1$作为特征值$\mathrm{一个}\left(吨\right)$, 写成$米(吨,-1)$. 证明了在所有树T中$p\ge 2$悬垂顶点，最大值$米(吨,-1)$是p -1，并且对于具有$p\ge 2$垂饰顶点，$米(吨,-1)=p-1$当且仅当$吨={磷}_n$和$n\equiv 2\left(\mathrm{米}\mathrm{○}\mathrm{d}3\right)$, 或T是一棵树，其中$d(v,你)\equiv 2\left(\mathrm{米}\mathrm{○}\mathrm{d}3\right)$对于T的任何下垂顶点v和任何主顶点u，其中主顶点是度数至少为 3 的顶点，并且$d(v,你)$是v和u之间的距离。