摘要

In [Linear Algebra Appl. 149 (1991) 19-34], Shao proved that for a tree T on n vertices, the kth eigenvalue lambda k(T) <= root[n/k]-1 (1 <= k <= [n/2]) and this bound is best possible for all n not equivalent to 0 (mod k), and for k = 1. In this note, we give a sufficient and necessary condition for the above equality, and the effect on the multiplicity of eigenvalues of two disjoint graphs by adding an edge between them is also considered.