On topological indices of random caterpillar graphs

We consider the asymptotic behaviour of topological indices of random caterpillar graphs, such as the Zagreb, $\alpha$-Randi\'c, Wiener, Gini and level indices, as the size of the graph grows. Our simple, general framework, which is based on a multivariate central limit theorem, the continuous mapping theorem and Slutsky's theorem, can also be used to cover other topological indices. We derive limit laws for these indices and provide exact and asymptotic results on their means and variances. The $\alpha$-Randi\'c index exhibits a discontinuity in its limit distribution at $\alpha=\frac{1}{2}$.