Distance Laplacian eigenvalues of graphs and chromatic and independence number

TL;DR

This paper investigates the eigenvalues of the distance Laplacian matrix of graphs, relating their distribution to graph invariants like chromatic number, independence number, and diameter, providing bounds and characterizations.

Contribution

It establishes new bounds on the number of eigenvalues in specific intervals based on graph invariants and characterizes graphs with certain eigenvalue properties.

Findings

Bound on eigenvalues in [n, n+2) related to chromatic number.

Bound on eigenvalues in [n, n+α(G)) related to independence number.

Characterization of diameter-2 graphs with specific eigenvalue counts.

Abstract

For a connected graph $G$ of order $n$, let $Diag(Tr)$ be the diagonal matrix of vertex transmissions and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as $D^L(G)=Diag(Tr)-D(G)$ and the eigenvalues of $D^{L}(G)$ are called the distance Laplacian eigenvalues of $G$. Let $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$ be the distance Laplacian eigenvalues of $G$. Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L} } I$) be the number of distance Laplacian eigenvalues of $G$ which lie in the interval $I$. For a prescribed interval $I$, we determine $m_{D^{L} }I$ in terms of independence number $\alpha(G)$, chromatic number $\chi$, number of pendant vertices and diameter $d$ of the graph $G$. In particular, we prove that $m_{D^{L}(G) }[n,n+2)\leq \chi-1$, ~$m_{D^{L}(G) }[n,n+\alpha(G))\leq n-\alpha(G)$ and we show that the inequalities are sharp. We also show that $m_{D^{L} (G )}\bigg( n,n+\left\lceil\frac{n}{\chi}\right\rceil\bigg)\leq n- \left\lceil\frac{n}{\chi}\right\rceil-C_{\overline{G}}+1 $, where $C_{\overline{G}}$ is the number of components in $\overline{G}$, and discuss some cases where the bound is best possible. In addition, we prove that $m_{D^{L} (G )}[n,n+p)\leq n-p$, where $p\geq 1$ is the number of pendant vertices. Also, we characterize graphs of diameter $d\leq 2$ which satisfy $m_{D^{L}(G) } (2n-1,2n )= \alpha(G)-1=\frac{n}{2}-1$. At the end, we propose some problems of interest.