Trees with extremal Laplacian eigenvalue multiplicity

Abstract

Let $T$ be a tree. Suppose $λ$ is an eigenvalue of the Laplacian matrix of $T$ with multiplicity $m_{T}(λ)$. It is known that $m_{T}(λ) \leq p(T)-1$, 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 $λ$ such that $m_{T}(λ)=p(T)-1$. We show that such trees are precisely either paths, or there exists an integer $q$ such that if $α$ and $β$ are two distinct pendant vertices, then the distance $d(α,β)$ satisfies $d(α, β) \equiv 2q ~{\rm{mod}}~(2q+1)$. As a consequence, we show that $1$ is an eigenvalue of $L_T$ with multiplicity $p(T)-1$ if and only if $d(α,β) \equiv 2\,\mbox{mod}\, 3$ for all distinct pendant vertices $α$ and $β$ of $T$.