Line graphs with the largest eigenvalue multiplicity

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 $m_{L(T)}(\lambda )\le p(T)-1$ for any tree $T$ with $p(T)\ge 3$, 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 $m_{L(G)}(\lambda )\le 2c(G)+p(G)-1$, 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.