A note on the multiplicities of graph eigenvalues

摘要

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_? at vertex i (i = 1,...,r, 1 ≤ r ≤ n). For a real symmetric matrix A whose graph is H, let m_A (μ) denote the multiplicity of an eigenvalue μ of A. From a result in da Fonseca (2005) [7], we know that m_A (μ)≤ n. In this note, we characterize the case m_A (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].