The eigenvalue multiplicity of line graphs

Abstract

Let m(G,λ), c(G) and p(G) be the multiplicity of an eigenvalue λ, the cyclomatic number and the number of pendant vertices of a connected graph G, respectively. Yang et al. (2023) [10] proved that m(L(T),λ)≤p(T)−1 for any tree T, and characterized all trees T with m(L(T),λ)=p(T)−1, where L(T) is the line graph of T. In this paper, we extend their result from a tree T to any graph G≠Cn, and prove that m(L(G),λ)≤2c(G)+p(G)−1 for any graph G≠Cn. Moreover, all graphs G with m(L(G),−1)=2c(G)+p(G)−1 are completely characterized.