Hosoya Polynomials Of Random Spiro And Polyphenyl Benzenoid Chains

Let G be a graph with vertex set V(G),dG(u,v) the distance between ver-tices u and v in G. The Hosoya polynomial H(G,x) of G is a polynomial in variable x. In this paper, we first give the recurrences or explicit formulae for computing the Hosoya polynomials of spiro and polyphenyl hexagonal chains, which are molecular graphs of two classes of unbranched mul-tispiro molecules and polycyclic aromatic hydrocarbons. Secondly, we show ex-plicit analytical expressions for the expected values of the Hosoya polynomials of a random spiro hexagonal chain and a random polyphenyl hexagonal chain with n hexagons. Finally, as corollaries, the expected values of a series of topological indices, such as Wiener indices, hyper-Wiener indices, Tratch-Stankevitch-Zefirov indices,of a random spiro hexagonal chain and a random polyphenyl hexagonal chain with n hexagons can be obtained from the expressions, to some extent, which generalize the results given by W. Yang et al.[Wiener index in random polyphenyl chains, MATCH Commun. Math. Comput. Chem.68(2012)371-376].