The number of Laplacian eigenvalues of trees less than one

Abstract

Let $m_T[0,1)$ denote the number of Laplacian eigenvalues of a tree $T$ that are strictly less than $1$. Guo, Xue and Liu \cite{GXL} proved that every tree $T$ of diameter $d$ satisfies $m_T[0,1)\ge \lceil (d+1)/3\rceil$, and that this bound is sharp when $d\equiv 2(\mathrm{mod}3)$. It is classical that every graph of diameter $d$ has domination number at least $\lceil (d+1)/3\rceil$. In this paper, for any diameter $d$, $m_T[0,1)=\lceil (d+1)/3\rceil$ if and only if $\gamma (T)=\lceil (d+1)/3\rceil$; we characterize all trees satisfying this necessary and sufficient condition, and also prove that almost all trees satisfy $m_T[0,1)\ge \lceil (d+1)/3\rceil +1$.

arXiv comment16 pages, 4 figures. Submitted to journal