Sharp lower bounds for the Laplacian Estrada index of graphs

Abstract

Let G be a simple graph on n vertices, and let λ1,λ2,…,λn be the Laplacian eigenvalues of G. The Laplacian Estrada index of G is defined as LEE(G)=∑i=1neλi. Consider a graph G with n≥3 vertices, m edges, c connected components, and the largest Laplacian eigenvalue λn. Let Kn, Sn, and Kp,q (p + q = n) denote the complete graph, the star graph, and the complete bipartite graph on n vertices, respectively. In this paper, we establish that LEE(G)≥ne2mn+c+eλn−(c+1)eλnc+1. Furthermore, we show that the equality holds if and only if G≅K¯n (the complement of Kn), G≅∪i=1c−1K1∪Sc+1 if n = 2c, or G≅Kn2,n2 if G is a connected graph on an even number of vertices. As a consequence of this lower bound, we derive sharp lower bounds for the Laplacian Estrada index of a graph, considering its well-known graph parameters. This leads to improvements to some previously known lower bounds for the Laplacian Estrada index of a graph. Notably, we establish a sharp lower bound for the Laplacian Estrada index of a graph in terms of its maximum vertex degree. As an application, we demonstrate that the lower bound for the Laplacian Estrada index presented by Khosravanirad in [A Lower Bound for Laplacian Estrada Index of a Graph, MATCH Commun Math Comput Chem. 2013;70:175–180.] is not complete. Consequently, we provide a complete version of this lower bound.