Trees with extremal Laplacian eigenvalue multiplicity

TL;DR

This paper characterizes trees with maximum Laplacian eigenvalue multiplicity, showing they are either paths or have a specific pendant vertex distance pattern, and identifies conditions for eigenvalue 1 multiplicity.

Contribution

It provides a complete characterization of trees where an eigenvalue's multiplicity reaches the upper bound, linking it to pendant vertex distances and path structures.

Findings

Trees with maximum eigenvalue multiplicity are either paths or have pendant vertices at distances satisfying a modular condition.

Eigenvalue 1 has multiplicity p(T)-1 if and only if all pendant vertices are separated by distances congruent to 2 mod 3.

The paper establishes a precise structural criterion for extremal Laplacian eigenvalue multiplicities in trees.

Abstract

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 $m_{T}(\lambda) \leq p(T)-1$, 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 $\lambda$ 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 $d(\alpha, \beta) \equiv 2q ~{\rm{mod}}~(2q+1)$. As a consequence, we show that $1$ is an eigenvalue of $L_T$ with multiplicity $p(T)-1$ if and only if $d(\alpha,\beta) \equiv 2\,\mbox{mod}\, 3$ for all distinct pendant vertices $\alpha$ and $\beta$ of $T$.