Trees with extremal Laplacian eigenvalue multiplicity

Let $T$ be a tree. Suppose $\lambda$ is an eigenvalue of the Laplacian matrix of $T$ with multiplicity $m_T(\lambda )$. It is known that $m_T(\lambda )\le p(T)-1$, where $p(T)$ is the number of pendant vertices of $T$. In this paper, we characterize all trees $T$ for which there exists an eigenvalue $\lambda$ such that $m_T(\lambda )=p(T)-1$. We show that such trees are precisely either paths, or there exists an integer $q$ such that if $\alpha$ and $\beta$ are two distinct pendant vertices, then the distance $d(\alpha ,\beta )$ satisfies $d(\alpha ,\beta )\equiv 2q\mathrm{m}\mathrm{o}\mathrm{d}(2q+1)$. As a consequence, we show that $1$ is an eigenvalue of $L_T$ with multiplicity $p(T)-1$ if and only if $d(\alpha ,\beta )\equiv 2mod3$ for all distinct pendant vertices $\alpha$ and $\beta$ of $T$.