Asymptotic values of four Laplacian-type energies for matrices with degree-distance-based entries of random graphs

Abstract

Let f(D(i,j),di,dj) be a real function symmetric in i and j with the property that f(d,(1+o(1))np,(1+o(1))np)=(1+o(1))f(d,np,np) for d=1,2. Let G be a graph, di denote the degree of a vertex i of G and D(i,j) denote the distance between vertices i and j in G. In this paper, we define the f-weighted Laplacian matrix for random graphs in the Erdös-Rényi random graph model Gn,p, where p∈(0,1) is fixed. Four weighted Laplacian type energies: the weighted Laplacian energy LEf(G), weighted signless Laplacian energy LEf+(G), weighted incidence energy IEf(G) and the weighted Laplacian-energy like invariant LELf(G) are introduced and studied. We obtain the asymptotic values of IEf(G) and LELf(G), and the values of LEf(G) and LEf+(G) under the condition that f(D(i,j),di,dj) is a function dependent only on D(i,j). As a consequence, we get that under the condition that f(D(i,j),di,dj) is a function dependent only on D(i,j), for almost all graphs Gp∈Gn,p, the energy for the matrix with degree-distance-based entries of Gp, E(Wf(Gp))<LEf(Gp), the Laplacian energy of the matrix, which can be viewed as a generalization of a conjecture by Gutman et al.