Remarks on the upper bound for the Randić energy of bipartite graphs

Abstract

Let G=(V,E), V={1,2,…,n} be a simple graph without isolated vertices, with n(n≥3) vertices and m edges, whose vertex degrees are given in the following form d1≥d2≥⋯≥dn>0. If A is the adjacency matrix, the Randić matrix R=‖Rij‖ is defined in the following way Rij={1didjifviandvjare adjacent ,0otherwise . The eigenvalues of matrix R, ρ1≥ρ2≥⋯≥ρn, are called the Randić eigenvalues of graph G. The Randić energy of graph G, denoted by RE, is defined in the following way: RE=RE(G)=∑i=1n|ρi|. In this paper, upper bounds for graph invariant RE have been studied.