Eigenvalue multiplicity of graphs with given cyclomatic number and given number of quasi-pendant vertices

Abstract

For a connected graph G, we denote by mG(μ), c(G) and p(G) the eigenvalue multiplicity of μ in G, the cyclomatic number and the number of pendant vertices in G, respectively. In 2020, Wang et al. (2020) proved that mG(μ)≤2c(G)+p(G) for any μ∈R, the equality holds if and only if G is a cycle and μ is a multiple eigenvalue of the cycle. We find that when ∥μ∥≥2, the bound 2c(G)+p(G) is much rough. Thus we are motivated to improve the bound of mG(μ) from 2c(G)+p(G) to c(G)+q(G) for the case when ∥μ∥≥2, where q(G) is the number of quasi-pendant vertices of G.