The asymptotic value of graph energy for random graphs with degree-based weights

Abstract

In this paper, we investigate the energy of a weighted random graph Gp(f) in Gn,p(f), in which each edge ij takes the weight f(di,dj), where dv is a random variable, the degree of vertex v in the random graph Gp of the Erdös–Rényi random graph model Gn,p, and f is a symmetric real function on two variables. Suppose |f(di,dj)|≤Cnm for some constants C,m>0, and f((1+o(1))np,(1+o(1))np)=(1+o(1))f(np,np). Then, for almost all graphs Gp(f) in Gn,p(f), the energy of Gp(f) is (1+o(1))f(np,np)83πp(1−p)⋅n3∕2, where p∈(0,1) is any fixed and independent of n. Consequently, with this one basket we can get the asymptotic values of various kinds of graph energies of chemical use, such as Randić energy, ABC energy, and energies of random matrices obtained from various kinds of degree-based chemical indices.