On the Laplacian Estrada Index of Graphs with Given Chromatic Number

Abstract

Assume that ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ are the eigenvalues of the Laplacian matrix of a graph $G$. The Laplacian Estrada index of $G$, denoted by $LEE(G)$, is defined as $LEE(G)=\sum _{i=1}^n{e}^{{\mu }_i}$. In this note, we give an upper bound on $LEE(G)$ in terms of chromatic number and characterize the corresponding extremal graph.