Hosoya Polynomials Of Two Classes Of Graphs

Let G be a simple graph and V(G) be the vertices set of G, the Hosoya polynomial of G is H(G,x)=Σ{u,v}(?)v(G)xdG(u,v) where dG(u,v) is the distance of u,v in G. Hosoya polynomial is introduced by Japanese scientist H. Hosoya in1988, and contains the distance distribution of all pairs the vertices of graph. In this paper, we discussed on the Hosoya polynomials of toroidal polyhex and phenylene chains, and obtained some result on distance-based topological indices.Toroidal polyhex Ⅱ(p, q, t) is a3—egular(or cubic)bipartite graph described by three parameters p, q and t, embedded on the torus such that each face is a hexagon. H. Zhang etc. have obtained the distances sequence and the formulae of Wiener index of toroidal polyhex under condition t=0or p≤2q or p≤q+t. Based on this result, we get the Hosoya polynomial of H(p,q,t) under the same condition. M. V. Diudea, M. Eliasi and B. Taeri have computed the Hosoya polynomial of toroidal polyhex under some condition respectively, which is indeed included in my conclusion. Further more, we get the Wiener index, hyper-Wiener index and TSZ index.Phenylene chains PH(L;Z) is consisting of hexagons and squares alternately, in which each square is aduacent to two hexagons, whereas the hexagons are not adjacent to each other. In this paper we decompose PH by mean of gated amalgamations, and then get the Hosoya polynomial of phenylene chains. Some relevant results are obtained.