A relation between multiplicity of nonzero eigenvalues of trees and their matching numbers

Abstract

Using star complement technique, Rowlinson [Linear Algebra Appl. 2010] proved that if a tree T, with diameter at least 4, has 1 as an eigenvalue of multiplicity k, then it has k+1 pendant edges that form an induced matching. In this paper, we apply the Parter-Wiener Theorem to generalize Rowlinson's result by replacing eigenvalue 1 with an arbitrary nonzero eigenvalue. Moreover, we describe the trees with a maximum matching or a maximum induced matching consisting of k+1 edges.