Distance Laplacian Eigenvalues and Chromatic Number in Graphs

Abstract

In the present paper we are interested in the study of the distance Laplacian eigenvalues of a connected graph with fixed order $n$ and chromatic number $\chi$. We prove lower bounds on the distance Laplacian spectral radius in terms of $n$ and $\chi$. We also prove results related to the distribution of the distance Laplacian eigenvalues with respect to the values of the chromatic number $\chi$. For some of the results, we characterize the extremal graphs, for others, we give examples of extremal graphs.