The Hosoya polynomial decomposition for catacondensed benzenoid graphs

For a graph G with the vertex set V(G), we denote by d(u,v) the distance between vertices u and v in G, by d(u) the degree of vertex u. The Hosoya polynomial of G is H(G)=∑_{u,v}⊆V(G)x^d(u,v). The partial Hosoya polynomials of G are for positive integer numbers m and n. It is shown that H(G₁)−H(G₂)=x²(x+1)²(H₃₃(G₁)−H₃₃(G₂)),H₂₂(G₁)−H₂₂(G₂)=(x²+x−1)²(H₃₃(G₁)−H₃₃(G₂)) and H₂₃(G₁)−H₂₃(G₂)=2(x²+x−1)(H₃₃(G₁)−H₃₃(G₂)) for arbitrary catacondensed benzenoid graphs G₁ and G₂ with equal number of hexagons. As an application, we give an affine relationship between H(G) with two other distance-based polynomials constructed by Gutman [I. Gutman, Some relations between distance-based polynomials of trees, Bulletin de l’Académie Serbe des Sciences et des Arts (Cl. Math. Natur.) 131 (2005) 1-7].