The multiplicity of eigenvalues of unicyclic graphs

Abstract

Let G be a graph on n vertices. Denote by mG,θ the multiplicity of θ as an eigenvalue of the adjacency matrix of G. In a recent paper, when G is a unicyclic graph, Tian and Wang showed that mG,θ≤⌊n−23⌋ when n≥11 and θ2 is an integer larger than 1. Motivated by it, we establish an upper bound of mG,θ for any fixed real number θ, generalizing the result of Tian and Wang. Some new techniques are developed to realize the generalization.