Hosoya Polynomials And Modified Hosoya Polynomials Of Chemistry Graphs

Let G=G(V, E) be a connected graph with vertex set V(G) and edge set E(G), dG(u, v) denotes the topological distance between vertices u and v in G, and δG(v) is the degree of the vertex v in G. The Hosoya polynomial H(G, x) of G is a polynomial in variable x, where the sum is taken over all unordered pairs of (not necessarily distinct) vertices in G. The Schultz polynomial of G is defined as and the modified Schultz polynomial of G is defined asIn this paper, we obtain explicit analytical expressions for expected values of Hosoya polynomial, Schultz polynomial and modified Schultz polynomial of a random benzenoid chain with n hexagons, and we investigate Hosoya polynomials of hexagonal trapeziums, tessellations of congruent regular hexagons shaped like trapeziums and give their explicit analytical expressions. As a special case, Hosoya polynomials of hexagonal triangles are obtained. Furthermore, as corollaries, the expected values of the well-known topological indices:Wiener index, Hyper-Wiener index, Tratch-Stankevitch-Zefirov index, Schultz index and modified Schultz index can be obtained by simple mathematical calculations.