完美匹配树最小正根的界

Bounds of the Smallest Positive Eigenvalue of a Tree with Perfecting Matching

设Ｔ为具有完美匹配的２ｑ阶树， ｑ（Ｔ）为其最小正根．又记Ｓｎ为Ｐｎ－１的一个邻接悬挂点接出一条新悬挂边而得的ｎ阶树．证明了：若 且 ，ｑ≥４，则 ．左边等式成立当且仅当 ．右边等式成立当且仅当 ．这里Ｐｑｑ为Ｐｑ的每个点都接出一条新悬挂边而得的２ｑ阶树， Ｔ＊１为Ｓ２ｑ－３的三个悬挂点各接出一条新悬挂边而得２ｑ阶树，而Ｔ＊２则是Ｓｑ的每个点都接出一条新悬挂边而得的２ｑ阶树．

Let T be a tree on 2q vertices with a perfect matching, q,(T) be the smallest positive eigenvalue of T. Denoted by Sn the tree with n vertices from the path Pn-1 by joining a new vertex to a neighboring end-vertex on Pn-1. In this paper, we demonstrate that if T P2q, and T Pqq, q≥4, q,(T*1)≤q,(T)≤q,(T*2), and the lower bound occurs only when T T*1 while the upper bound occurs only when T T*2, where Pqq is the tree from Pq by joining a new vertex to each vertex in Pn; T*1 is the tree from S2q-3by joining a new vertex to each end -vertex in S2q-3; and T*2 is the tree from Sq, by joining a new vertex to each vertex in Sq.