A note on the kth eigenvalue of trees

In [Linear Algebra Appl. 149 (1991) 19–34], Shao proved that for a tree T on n vertices, the kth eigenvalueλk(T)⩽nk-11⩽k⩽n2and this bound is best possible for all n ≢ 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.