Laplacian eigenvalue distribution, diameter and domination number of trees

Abstract

For a graph $G$ with domination number $\gamma$, Hedetniemi, Jacobs and Trevisan [European Journal of Combinatorics 53 (2016) 66-71] proved that $m_{G}[0,1)\leq \gamma$, where $m_{G}[0,1)$ means the number of Laplacian eigenvalues of $G$ in the interval $[0,1)$. Let $T$ be a tree with diameter $d$. In this paper, we show that $m_{T}[0,1)\geq (d+1)/3$. However, such a lower bound is false for general graphs. All trees achieving the lower bound are completely characterized. Moreover, for a tree $T$, we establish a relation between the Laplacian eigenvalues, the diameter and the domination number by showing that the domination number of $T$ is equal to $(d+1)/3$ if and only if it has exactly $(d+1)/3$ Laplacian eigenvalues less than one. As an application, it also provides a new type of trees, which show the sharpness of an inequality due to Hedetniemi, Jacobs and Trevisan.