A characterization on trees $T$ with $m(T,\lambda )=p(T)-2$

Let $m(G,\lambda )$ be the multiplicity of an eigenvalue $\lambda$ of a connected graph $G$. Wang et al. [Linear Algebra Appl. 584(2020), 257-266] proved that for any connected graph $G\mathrm{\ne }{C}_n$, $m(G,\lambda )\le 2c(G)+p(G)-1$, where $c(G)=\mathrm{\mid }E(G)\mathrm{\mid }-\mathrm{\mid }V(G)\mathrm{\mid }+1$ and $p(G)$ are the cyclomatic number and the number of pendant vertices of $G$, respectively. In the same paper, they proposed the problem to characterize all connected graphs $G$ with eigenvalue $\lambda$ such that $m(G,\lambda )=2c(G)+p(G)-1$. Wong et al. [Discrete Math. 347(2024), 113845] solved this problem for the case when $G$ is a tree by characterizing all trees $T$ with eigenvalue $\lambda$ such that $m(T,\lambda )=p(T)-1$. In this paper, we further provide the structural characterization on trees $T$ with eigenvalue $\lambda$ such that $m(T,\lambda )=p(T)-2$.