Enumeration of Wiener indices in random polygonal chains

The Wiener index of a connected graph (molecule graph) G is the sum of the distances between all pairs of vertices of G, which was reported by Wiener in 1947 and is the oldest topological index related to molecular branching. In this paper, simple formulae of the expected value of the Wiener index in a random polygonal chain and the asymptotic behavior of this expectation are established by solving a difference equation. Based on the results above, we obtain the average value of the Wiener index of all polygonal chains with n polygons. As applications, we use the unified formulae to obtain the expected values of the Wiener indices of some special random polygonal chains which were deeply discussed in the context of organic chemistry or statistical physics.