Asymptotic Distribution of Degree-Based Topological Indices

Topological indices play a significant role in mathematical chemistry. Given a graph G with vertex set V = {1, 2,..., n} and edge set epsilon, let d(i) be the degree of node i. The degree-based topological index is defined as I-n = Sigma({i,j}is an element of epsilon) f(d(i), d(j)), where f(x, y) is a symmetric function. In this paper, we investigate the asymptotic distribution of the degree-based topological indices of a heterogeneous Erdos-Renyi random graph. We show that after suitably centered and scaled, the topological indices converges in distribution to the standard normal distribution. Interestingly, we find that the general Randic index with f(x, y) = (xy)(tau) for a constant t exhibits a phase change at tau = -1/2.