Line graphs with the largest eigenvalue multiplicity

TL;DR

This paper fully characterizes graphs whose line graphs have an eigenvalue multiplicity equal to twice their cyclomatic number plus the number of pendant vertices minus one, extending previous results on trees and non-cycle graphs.

Contribution

It provides a complete characterization of all graphs with maximum eigenvalue multiplicity in their line graph for any eigenvalue, solving an open problem in spectral graph theory.

Findings

Characterization of graphs with maximum eigenvalue multiplicity in line graphs.

Extension of previous bounds to arbitrary eigenvalues.

Resolution of an open problem in spectral graph theory.

Abstract

For a connected graph $G$, we denote by $L(G)$, $m_{G}(\lambda)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $\lambda$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In 2023, Yang et al. \cite{WL LT} proved that $m_{L(T)}(\lambda)\leq p(T)-1$ for any tree $T$ with $p(T)\geq 3$, and characterized all trees $T$ with $m_{L(T)}(\lambda) = p(T)-1$. In 2024, Chang et al. \cite{-1 LG} proved that, if $G$ is not a cycle, then $m_{L(G)}(\lambda)\leq 2c(G)+p(G)-1$, and characterized all graphs $G$ with $m_{L(G)}(-1) = 2c(G)+p(G)-1$. The remaining ploblem is to characterize all graphs $G$ with $m_{L(G)}(\lambda)= 2c(G)+p(G)-1$ for an arbitrary eigenvalue $\lambda$ of $L(G)$. In this paper, we give this problem a complete solution.