Laplacian eigenvalue distribution, diameter and domination number of trees

TL;DR

This paper investigates the relationship between Laplacian eigenvalues, diameter, and domination number in trees, establishing new bounds and characterizations that deepen understanding of their spectral and structural properties.

Contribution

It provides a lower bound on Laplacian eigenvalues in trees based on diameter, characterizes trees achieving this bound, and links domination number with eigenvalues and diameter.

Findings

Lower bound of (d+1)/3 for Laplacian eigenvalues in trees

Complete characterization of trees reaching the lower bound

Establishment of a relation between domination number, diameter, and eigenvalues

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.