The Hosoya polynomial decomposition for hexagonal chains

For a graph GG we denote by dG(u,v)dG(u,v) the distance between vertices uu and vv in GG, by dG(u)dG(u) the degree of vertex uu. The Hosoya polynomial of GG is H(G)=∑{u,v}⊆V(G)xdG(u,v)H(G)=∑{u,v}⊆V(G)xdG(u,v). For any positive numbers mm and nn, the partial Hosoya polynomials of GG are Hm(G)=∑{u,v}⊆V(G)dG(u)=dG(v)=mxdG(u,v), Hmn(G)=∑{u,v}⊆V(G)dG(u)=m,dG(v)=nxdG(u,v). It has been shown that H(G1)−H(G2)=x2(x+1)2(H3(G1)−H3(G2)),H2(G1)−H2(G2)=(x2+x−1)2(H3(G1)−H3(G2))H(G1)−H(G2)=x2(x+1)2(H3(G1)−H3(G2)),H2(G1)−H2(G2)=(x2+x−1)2(H3(G1)−H3(G2)) and H23(G1)−H23(G2)=2(x2+x−1)(H3(G1)−H3(G2))H23(G1)−H23(G2)=2(x2+x−1)(H3(G1)−H3(G2)) for arbitrary hexagonal chains G1G1 and G2G2 with the same number of hexagons. As a corollary, we give an affine relationship between H(G)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].