A characterization of graphs $G$ with $m_G(\lambda )=2c(G)+q_s(G)-1$

Let $G$ be a simple connected graph. If every pendant path in $G$ is at least $P_s$, we denote that $G\in {\mathbb{G}}_s$. For $G\in {\mathbb{G}}_s$, let $Q_s(G)$ be the set of vertices in $G$ that are distance $s$ from the pendant vertex, and let $\mathrm{\mid }{Q}_s(G)\mathrm{\mid }=q_s(G)$. For $G\in {\mathbb{G}}_s$, Li et al. (2024) proved that when $\lambda$ is not an eigenvalue of $P_s$ and $G$ is neither a cycle nor a starlike tree $T_k$, it holds that $m_G(\lambda )\le 2c(G)+q_s(G)-1$ and characterized the extremal graphs when $G$ is a tree. In this article, we characterize the extremal graphs for which $m_G(\lambda )=2c(G)+q_s(G)-1$ when $G\in {\mathbb{G}}_s$ and $\lambda \mathrm{\notin }\sigma (P_s)$.