%0 Journal Article
%T On the k th Eigenvalues of Trees with Perfect Matchings
%A Wai Chee Shiu
%A An Chang
%J Discrete Mathematics & Theoretical Computer Science
%D 2007
%I Discrete Mathematics & Theoretical Computer Science
%X Let 有 + 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 有 + 2p, its k-th largest eigenvalue 竹 k (T) satisfies 竹 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 有 * 2p. We also prove that for any tree T in 有 * 2p, its k-th largest eigenvalue 竹 k (T) satisfies 竹 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 {竹} k * (2p)> 1 / 2(﹟{t-1-﹟{(k-1)/(t-k)}}+ ﹟{t+3-﹟{(k-1)/(t-k)}}) t=  p / k  , where {竹} k ^* (2p) is the maximum value of the k-th largest eigenvalue of the trees in 有 * 2p. By this inequality, it is easy to see that the above upper bound on 竹 k (T) for T﹋ 有 * 2p turns out to be asymptotically good when p√ 0mod k.
%U http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/484