Trees with extremal Laplacian eigenvalue multiplicity

Abstract

Let $T$ be a tree. Suppose $\hat{I}»$ is an eigenvalue of the Laplacian matrix of $T$ with multiplicity $m_T(\hat{I}»)$. It is known that $m_T(\hat{I}»)\le p(T)-1$, 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 $\hat{I}»$ such that $m_T(\hat{I}»)=p(T)-1$. We show that such trees are precisely either paths, or there exists an integer $q$ such that if $\hat{I}\pm$ and ${\hat{I}}^2$ are two distinct pendant vertices, then the distance $d(\hat{I}\pm ,{\hat{I}}^2)$ satisfies $d(\hat{I}\pm ,{\hat{I}}^2)\equiv 2q\mathrm{mod}(2q+1)$. 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 $\hat{I}\pm$ and ${\hat{I}}^2$ of $T$.

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