Hosoya Polynomials Of TUC₄C₈(R) Nanotubes

For a connected graph G, the Hosoya polynomial (or called Wiener polynomial) of G is a distance-based polynomial and is defined as: H(G, x) =(?) x^d(u,v),where V(G) is the set of all vertices of G and d(u, v) is the distance between vertices u and v. It has the property that its first derivative at x = 1 is equal to the Wiener index. The Wiener index of G is the sum of distances between all pairs of vertices. It is a well-known topological index based on the distances and has found extensive application in chemistry. In this paper, we shall compute the Hosoya polynomials of TUC₄C₈(R) nanotubes, covering a nanotube by alternating rhombs C₄ and octagons C₈. we first obtain the distances from all vertices of every rhombs to some special vertices. Then, by proposing a recursive method, we obtain the analytical expressions for Hosoya polynomials of TUC₄C₈(R) nanotubes. Furthermore, the Wiener index and the hyper-Wiener index can be calculated.