On the k th Eigenvalues of Trees with Perfect Matchings

Let Τ + 2p be the set of all trees on 2p (p≥ 1) vertices with perfect matchings. In this paper, we prove that for any tree T in Τ + 2p, its k-th largest eigenvalue λ k (T) satisfies λ k (T)≤ 1 / 2 (√{ p / k -1}+ √{ p / k +3}) (k=1,2,..,p) and show that this upper bound is the best possible when k=1. The set of trees obtained from a tree on p vertices by joining a pendent vertex to each vertex of the tree, respectively, is denoted by Τ * 2p. We also prove that for any tree T in Τ * 2p, its k-th largest eigenvalue λ k (T) satisfies λ k (T)≤ 1 / 2 (√{ p / k -1}+√{ p / k +3}) (k=1,2,…,p) and show that this upper bound is the best possible when k=1 or p ≡ 0 mod k. We further give the following inequality {λ} k * (2p)> 1 / 2(√{t-1-√{(k-1)/(t-k)}}+ √{t+3-√{(k-1)/(t-k)}}) t= p / k , where {λ} k ^* (2p) is the maximum value of the k-th largest eigenvalue of the trees in Τ * 2p. By this inequality, it is easy to see that the above upper bound on λ k (T) for T∈ Τ * 2p turns out to be asymptotically good when p≡ 0mod k.