## Hosoya polynomials of the capped zig-zag nanotubes

##### Abstract

The Hosoya polynomial of a graph G is defined as H (G, x) = Sigma(k >= 0) d(G, k)x(k), where d(G, k) is the number of the vertex pairs at distance k in G. The calculation of Hosoya polynomials of molecular graphs is a significant topic because some important molecular topological indices such as Wiener index, hyper-Wiener index, and Wiener vector, can be obtained from Hosoya polynomials. Hosoya polynomials of zig-zag open-ended nanotubes have been given by Xu and Zhang et al. A capped zig-zag nanotube T(p,q)[C, D; a] consists of a zig-zag open-ended nanotube T (p, q) and two caps C and D with the relative position a between C and D. In this paper, we give a general formula for calculating Hosoya polynomial of any capped zig-zag nanotube. By the formula, Hosoya polynomial of any capped zig-zag nanotube can be deduced. Furthermore, it is also shown that any two non-isomorphic capped zig-zag nanotube T(p, q)[C, D; a(1)], T (p, q)[C, D; a(2)] with q >= q* >= p + 1 have the same Hosoya polynomial, where q* is a integer which depends on structures of C and D.