### On the k^{th} Eigenvalues of Trees with Perfect Matchings

*Wai Chee Shiu, An Chang*

#### Abstract

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,&elip;,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
⌋,

_{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.Full Text: PDF PostScript