The Hosoya polynomial decomposition for hexagonal chains

Abstract

For a graph $G$ we denote by $d_G(u,v)$ the distance between vertices $u$ and $v$ in $G$, by $d_G\left(u\right)$ the degree of vertex $u$. The Hosoya polynomial of $G$ is $H\left(G\right)={\sum }_{\{u,v\}\subseteq V\left(G\right)}{x}^{d_G(u,v)}$. For any positive numbers $m$ and $n$, the partial Hosoya polynomials of $G$ are $H_m\left(G\right)={\sum }_{\begin{array}{c}\hfill \{u,v\}\subseteq V\left(G\right)\hfill \\ \hfill d_G\left(u\right)=d_G\left(v\right)=m\hfill \end{array}}{x}^{d_G(u,v)}$, $H_{mn}\left(G\right)={\sum }_{\begin{array}{c}\hfill \{u,v\}\subseteq V\left(G\right)\hfill \\ \hfill d_G\left(u\right)=m,d_G\left(v\right)=n\hfill \end{array}}{x}^{d_G(u,v)}$. It has been shown that $H\left(G_1\right)-H\left(G_2\right)=x^2{(x+1)}^2(H_3\left(G_1\right)-H_3\left(G_2\right)),H_2\left(G_1\right)-H_2\left(G_2\right)={(x^2+x-1)}^2(H_3\left(G_1\right)-H_3\left(G_2\right))$ and $H_{23}\left(G_1\right)-H_{23}\left(G_2\right)=2(x^2+x-1)(H_3\left(G_1\right)-H_3\left(G_2\right))$ for arbitrary hexagonal chains $G_1$ and $G_2$ with the same number of hexagons. As a corollary, we give an affine relationship between $H\left(G\right)$ 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].