Hosoya polynomial and topological indices of n-linear benzene

The  Hosoya  polynomial  was  introduced  by  Hosoya  in  1988  for  a  molecular  graph  G  as H (G, x) = ∑^d(G)  d (G,k) x^k    where d(G, k) is the number of pairs of vertices of G laying at 

                       ^k-1

distance k from each other to count the number of paths of different lengths in G. The most interesting application of the Hosoya polynomial is that almost all distance-based topological indices can be recovered from it. In this article, we give the general closed form of the Hosoya polynomial of n times linearly concatenated benzene molecule B_n by partitioning the vertices of B_n and by using induction to find sums of distances of different lengths in B_n. We also find Wiener, hyper Wiener, Harary, and TSZ indices to predict physical, chemical and pharmacological properties of B_n and molecules containing B_n.