The eigenvalue multiplicity of line graphs

Abstract

Let $m(G,\lambda )$, $c\left(G\right)$ and $p\left(G\right)$ 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 $m\left(L\right(T),\lambda )\le p\left(T\right)-1$ for any tree T, and characterized all trees T with $m\left(L\right(T),\lambda )=p\left(T\right)-1$, where $L\left(T\right)$ is the line graph of T. In this paper, we extend their result from a tree T to any graph $G\ne C_n$, and prove that $m\left(L\right(G),\lambda )\le 2c\left(G\right)+p\left(G\right)-1$ for any graph $G\ne C_n$. Moreover, all graphs G with $m\left(L\right(G),-1)=2c\left(G\right)+p\left(G\right)-1$ are completely characterized.