  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 $\tau=-\frac{1}{2}$.
