On the k(th) eigenvalues of trees with perfect matchings
Let T-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 T-2p(+), the kth largest eigenvalue lambda(k)(T) satisfies lambda(k)(T) <= 1/2 (root [p/k] - 1 + root [p/k] +3) (k = 1, 2,...,p). This upper bound is known to be 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 is denoted by T-2p(*). We also prove that for any tree T in T-2p(*), its kth largest eigenvalue lambda(k)(T) satisfies lambda(k)(T) <= 1/2 (root [p/k] - 1 + root [p/k] + 3) (k = 1; 2,...,p) and show that this upper bound is best possible when k = 1 or p not equivalent to 0 (mod k). We further give the following inequality [GRAPHICS] where lambda(*)(k)(2p) is the maximum value of the kth largest eigenvalue of the trees in T-2p(*). By this inequality, it is easy to see that the above upper bound on lambda(k)(T) for T is an element of T-2p(*) turns out to be asymptotically tight when p equivalent to 0 (mod k).
eigenvalue tree perfect matching