Relationship between the Hosoya polynomial and the hyper-Wiener index Original Research Article

The Hosoya polynomial of a graph, H(G, x), has the property that its first derivative, evaluated at x = 1, equals the Wiener index, i.e., W(G) = H′(G, 1). In this paper, an equation is presented that gives the hyper-Wiener index, WW(G), in terms of the first and second derivatives of H(G, x). Also defined here is a hyper-Hosoya polynomial, HH(G, x), which has the property WW(G) = HH′(G, 1), analogous to W(G) = H′(G, 1). Uses of higher derivatives of HH(G, x) are proposed, analogous to published uses of higher derivatives of H(G, x).