The Hosoya polynomial decomposition for catacondensed benzenoid graphs

For a graph GG with the vertex set V(G)V(G), we denote by d(u,v)d(u,v) the distance between vertices uu and vv in GG, by d(u)d(u) the degree of vertex uu. The Hosoya polynomial of GG is H(G)=∑{u,v}⊆V(G)xd(u,v)H(G)=∑{u,v}⊆V(G)xd(u,v). The partial Hosoya polynomials of GG are Hmn(G)=∑{u,v}⊆V(G)d(u)=m,d(v)=nxd(u,v) for positive integer numbers mm and nn. 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))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))H23(G1)−H23(G2)=2(x2+x−1)(H33(G1)−H33(G2)) for arbitrary catacondensed benzenoid graphs G1G1 and G2G2 with equal number of hexagons. As an application, we give an affine relationship between H(G)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].