Abstract
Topological indices play a significant role in mathematical chemistry. Given a graph \( \mathcal{G} \) with vertex set \( \mathcal{V}=\{1,2,\dots,n\} \) and edge set \( \mathcal{E} \), let \( d_i \) be the degree of node \( i \). The degree-based topological index is defined as \( \mathcal{I}_n= \sum_{\{i,j\}\in \mathcal{E}}f(d_i,d_j) \), 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 Erdős-Rényi 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 Randi\'{c} index with \( f(x,y)=(xy)^{\tau} \) for a constant \( \tau \) exhibits a phase change at \( \tau=-\frac{1}{2} \).
