Distance Laplacian eigenvalues of graphs, and chromatic and independence number

Abstract

Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L}} I$) be the number of distance Laplacian eigenvalues of a graph $G$ which lie in $I$. For a prescribed interval $I$, we give the bounds for $m_{D^{L} }I$ in terms of the independence number $\alpha(G)$, the chromatic number $\chi$, the number of pendant vertices $p$, the number of components in the complement graph $C_{\overline{G}}$ and the diameter $d$ of $G$. In particular, we prove that $m_{D^{L}(G) }[n,n+2)\leq \chi-1$, $m_{D^{L}(G)}[n,n+\alpha(G))\leq n-\alpha(G)$, $m_{D^{L}(G) }[n,n+p)\leq n-p$ and discuss the cases where the bounds are best possible. In addition, we characterize graphs of diameter $d\leq 2$ which satisfy $m_{D^{L}(G) } (2n-1,2n )= \alpha(G)-1=\frac{n}{2}-1$. We also propose some problems of interest.