Asymptotic distribution of degree--based topological indices

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 Erd\H{o}s-R\'{e}nyi 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 $\tau=-\frac{1}{2}$.