A note on the multiplicities of graph eigenvalues

Let G   be a graph with vertex set {1,…,n}{1,…,n}, and let H   be the graph obtained by attaching one pendant path of length kiki at vertex i (i=1,…,r,1≤r≤n)i (i=1,…,r,1≤r≤n). For a real symmetric matrix A whose graph is H  , let mA(μ)mA(μ) denote the multiplicity of an eigenvalue μ of A. From a result in da Fonseca (2005) [7], we know that mA(μ)≤nmA(μ)≤n. In this note, we characterize the case mA(μ)=nmA(μ)=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].