Distance Laplacian eigenvalues of graphs and chromatic and independence number

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 ${\partial }_1^L(G)\ge {\partial }_2^L(G)\ge \cdots \ge {\partial }_n^L(G)$ 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 $m_{D^L(G)}[n,n+2)\le \chi -1$, ~$m_{D^L(G)}[n,n+\alpha (G))\le n-\alpha (G)$ and we show that the inequalities are sharp. We also show that $m_{D^L(G)}(n,n+\lceil \frac{n}{\chi }\rceil )\le n-\lceil \frac{n}{\chi }\rceil -C_{\overline{G}}+1$, 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 $m_{D^L(G)}[n,n+p)\le n-p$, where $p\ge 1$ is the number of pendant vertices. Also, we characterize graphs of diameter $d\le 2$ 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.

17 pages