Eigenvalue multiplicity of a graph in terms of the number of external vertices

Abstract

The multiplicity of an eigenvalue λ of a graph G is denoted by m(G,λ). In a connected graph G with at least two vertices, a vertex is called external if it is not a cut vertex. In a tree, an external vertex is exactly a pendant vertex. Let ϵ(G) be the number of external vertices in G. In this paper, we prove that m(G,λ)≤ϵ(G)−1 for any λ and characterize the extremal graphs with m(G,−1)=ϵ(G)−1, which generalizes the main result of Wang et al. [Linear Multi. Alg., 2020] from a tree to an arbitrary connected graph.