Lower bounds for the Laplacian energy of bipartite graphs

Abstract

Given a simple undirected graph G=(V,E) with n vertices, if for the largest eigenvalue of its Laplacian matrix λ1 there exists a lower bound λ1≥α≥dGnn−1, then we have that its Laplacian energy satisfies LE(G)≥max{2dG,2(α−dG)},where dG=d1+⋯dnn is the average degree of G. This generic lower bound, obtained with the majorization technique, allows us to obtain two lower bounds for LE(G) which are valid for any connected bipartite graph, and for which the equalities are attained by Kn2,n2 and Sn.