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

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