The Hosoya polynomial decomposition for hexagonal chains

For a graph G we denote by dG(u, v) the distance between vertices u and v in G, by dG(u) the degree of vertex u. The Hosoya polynomial of G is H(G) = ∑ {u,v}⊆V (G) x dG (u,v). For any positive numbers m and n, the partial Hosoya polynomials of G are Hm(G) = ∑ {u, v} ⊆ V (G) dG (u) = dG (v) = m xdG (u,v), Hmn(G) = ∑ {u, v} ⊆ V (G) dG (u) = m, dG (v) = n xdG (u,v). It has been shown that H(G1) − H(G2) = x 2(x + 1)(H3(G1) − H3(G2)), H2(G1) − H2(G2) = (x 2 + x − 1)(H3(G1) − H3(G2)) and H23(G1) − H23(G2) = 2(x 2 + x − 1)(H3(G1) − H3(G2)) for arbitrary hexagonal chains G1 and G2 with the same number of hexagons. As a corollary, we give an affine relationship between H(G) and other two distance-based polynomials constructed by Gutman [I. Gutman, Some relations between distance-based polynomials of trees, Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.) 131 (2005) 1–7]. c © 2007 Elsevier Ltd. All rights reserved.