森的第k个特征根的Sharp下界

Sharp Lower Bound on the kth Largest Eigenvalues of Forest

设λ_k是n个顶点森的第k个特征根,q是边独立数。本文证明了关于λ_k（2≤k≤q-1）的一个猜想,同时给出了λ_k的下界;并且关于森获得了λ_k的Sharp下界,关于树在k较小时获得了λ_k的Sharp下界。

Let λh be the kth largest eigenvalue of a tree T (or a forest F) with n vertices, i.e., λ1≥λ2≥…≥λn, then λh= -λn-h+1, and λq is the smallest positive eigenvalue where q is the edge independence number. A conjecture about the lower bound of λh was proposed as follows:If k≤q, then λk(T)≥λk(Sn-2k+22k-2) with an equality iff T Sn-2k+22k-2, where Sn-2k+22k-2 is a tree formed by connecting with an edge a vertex of degree one of P2k-2 to the center of K1,n-2k+1. The following results bave been obtained:a. If k<q or k = q(under certain conditions), then the conjecture holds;b. λk(F)≥2cos[(k-1)π/(2k-1)], k = 2,…,q-1;c. The sharp lower bound of the kth (k≥1) positive eigenvalues of forest is 2cos [kπ/(2k + 1)];d. The sharp lower bound of the kth (k = 2,3,4,5) positive eigenvalues of trees is 2cosθk, where θk is The unique solution to sin(2k + 1)θ- (n -2k) ·sin(2k-1)θ=0 on ((k-1)π/(2k -1), kπ/(2k +1)].