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 $G_p(f)$ in $G_{n,p}(f)$, in which each edge $ij$ takes the weight $f(d_i,d_j)$, where $d_v$ is a random variable, the degree of vertex $v$ in the random graph $G_p$ of the Erd\"{o}s--R\'{e}nyi random graph model $G_{n,p}$, and $f$ is a symmetric real function on two variables. Suppose $|f(d_i,d_j)|\leq C n^m$ 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 $G_p(f)$ in $G_{n,p}(f)$, the energy of $G_p(f)$ is $(1+o(1))f(np,np)\frac{8}{3\pi}\sqrt{p(1-p)}\cdot n^{3/2},$ where $p\in(0,1)$ 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 Randi\'c energy, ABC energy, and energies of random matrices obtained from various kinds of degree-based chemical indices.