Graphs with maximal Laplacian eigenvalue multiplicity

In this paper, $G$ is a simple connected graph, and $m_G(\lambda)$ denotes the multiplicity of $\lambda$ as an eigenvalue of the Laplacian matrix $L(G)$. Let $p(G)$ denote the number of pendant vertices of $G$, $q(G)$ the number of quasi-pendant vertices of $G$, and $c(G)$ the dimension of the cycle space of $G$. Li et al. [Discrete Mathematics, 2026] proved that if $G$ is a tree with $G \not\cong K_{1,n-1}$ and $\lambda \neq 1$, then $$m_G(\lambda) \le q(T) - 1.$$ Moreover, for a general graph $G$ with $\lambda \neq 1$, Li et al. also proved in the same paper that $$m_G(\lambda) \le c(G) + q(G),$$ with equality if and only if $G \cong K_{1,n-1}$ or $G \cong C_n$ with $\lambda \notin \{0, 4\}$. A natural consequence is that if $c(G) + q(G) \ge 2$, then $$m_G(\lambda) \le 2c(G) + q(G) - 1.$$ When $c(G) = 0$, this reduces to the result of Li et al. for trees. In this paper, we give a complete characterization of graphs $G$ attaining the equality $$m_G(\lambda) = 2c(G) + q(G) - 1.$$