Wiener Indices And Hosoya Polynomials Of Graphs

The Wiener index of a graph is the sum of distances between all pairs of vertices. A toroidal polyhex (or toroidal graphitoid) is a 3-regular bipartite graph embedded on the torus such that each face is bounded by a hexagon and can be described by a string (p, q, t) of three integers (p≥1,q≥1,0≤t≤p - 1). In a recent work [MATCH 45 (2002) 109-122] M.V. Diudea obtained Wiener index formulae for several classes of toroidal nets, including toroidal polyhex with t = -q/2(mod p). In Chapter 2, we obtain formulae for calculating the Wiener index of toroidal polyhex with either t = 0 or p≤2q or p≤q + t, including t (?) -q/2 (mod p).For a connected graph G we denote by d(G, k) the number of pairs of vertices at distance k. The Hosoya polynomial of G is H(G,x) In Chapter 3, we give analyticalformulae for calculating the polynomials of zig-zag open-ended nanotubes and armchair open-ended nanotubes. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper-Wiener index, from a half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. In addition, we show the Hosoya polynomials of zigzag open-ended nanotubes are unimodal.A subgraph H of a graph G is gated if for every vertex x outside H there exists a vertex x' in H such that each vertex y of H is connected with a; by a shortest path passing through x'. The gated amalgam of graphs G₁ and G₂ is obtained from G₁ and G₂ by identifying their isomorphic gated subgraphs H₁ and H₂. In Chapter 4, two theorems on recurrence relations of Hosoya polynomials of gated amalgams are provided. As their applications, explicit expressions for the Hosoya polynomials of hexagonal chains are obtained.Similar to the well-known Wiener index, Eu et al. [Int. J. Quantum Chem. 106 (2006) 423-435] introduced three families of topological indices including Schultz index and modified Schultz index, called generalized Wiener indices, and then gave the similar formulae of generalized Wiener indices of hexagonal chains. They also mentioned three families of counting polynomials in x, called generalized Hosoya polynomials in contrast to the (standard) Hosoya polynomial, such that their first derivatives at x = 1 are equal to generalized Wiener indices. In Chapter 5, we gave explicit analytical expressions for generalized Hosoya polynomials of hexagonal chains.For a chemical graph G we denote by d_G (u, v) the distance between vertices u and v in G, by d_G(u) the degree (valence) of vertex u. The definition of Hosoya polynomial of G in an alternative way is For any positive integer num-bers m and n, the partial Hosoya polynomials of G are H_mn(G)= In particular, let H_m(G) = H_mm(G). In Chapter 6, we show that H(G₁)= H(G₂) = x²(x + 1)²(H₃(G₁) - H₃(G₂)), H₂(G₁) - H₂(G₂) = (x² + x - 1)²(H₃(G₁) - H₃(G₂)) and H₂₃(G₁) -H₂₃(G₂) = 2(x²+x -1) (H₃(G₁)—H₃(G₂)) for arbitrary catacondensed benzenoid graphs G₁ and G₂ with equal number of hexagons. As a corollary, we give an affine relationship between H(G) and other two distance-based polynomials constructed by Gutman [Bulletin de l'Academie Serbe des Sciences et des Arts 131 (2005) 1-7].