Line graphs with the largest eigenvalue multiplicity

For a connected graph G, we denote by $L\left(G\right)$, $m_G\left(\lambda \right)$, $c\left(G\right)$ and $p\left(G\right)$ the line graph of G, the eigenvalue multiplicity of λ in G, the cyclomatic number and the number of pendant vertices in G, respectively. In 2023, Yang et al. [12] proved that $m_{L\left(T\right)}\left(\lambda \right)\le p\left(T\right)-1$ for any tree T with $p\left(T\right)\ge 3$, and characterized all trees T with $m_{L\left(T\right)}\left(\lambda \right)=p\left(T\right)-1$. In 2024, Chang et al. [2] proved that, if G is not a cycle, then $m_{L\left(G\right)}\left(\lambda \right)\le 2c\left(G\right)+p\left(G\right)-1$, and they characterized all graphs G with $m_{L\left(G\right)}(-1)=2c\left(G\right)+p\left(G\right)-1$. The authors of [2] particularly stated that it seems somewhat difficult to characterize the extremal graphs G with $m_{L\left(G\right)}\left(\lambda \right)=2c\left(G\right)+p\left(G\right)-1$ for an arbitrary eigenvalue λ of $L\left(G\right)$. In this paper, we give this problem a complete solution.