A note on the multiplicities of graph eigenvalues

Let G be a graph with vertex set {1, ..., n}, and let H be the graph obtained by attaching one pendant path of length k(i) at vertex i (i = 1, ..., r, 1 <= r <= n). For a real symmetric matrix A whose graph is H, let m(A)(mu) denote the multiplicity of an eigenvalue mu of A. From a result in da Fonseca (2005) [7], we know that m(A)(mu) <= n. In this note, we characterize the case m(A)(mu) = n. We also give two upper bounds on eigenvalue multiplicity of trees and unicyclic graphs, which are generalizations of. some results in Rowlinson (2010) [10]. (C) 2013 Elsevier Inc. All rights reserved.