Wiener, edge-Wiener, and vertex-edge-Wiener index of Basilica graphs

We determine the exact value of the Wiener index, the edge-Wiener index, and the vertex-edge-Wiener index of the Basilica graphs, i.e., the sequence of finite Schreier graphs associated with the action of the Basilica group on the rooted binary tree. Moreover, we give a formula for the total distance of every vertex in the Basilica graphs, and we are able to make it explicit for some special vertices. We finally introduce the notions of asymptotic Wiener index and asymptotic total distance, which are compatible with that of convergence of the sequence of finite Basilica graphs to an infinite orbital limit graph in the Gromov–Hausdorff topology: the asymptotic values are explicitly computed.