The Hosoya polynomial decomposition for hexagonal chains

For a graph G we denote by d G ( u , v ) the distance between vertices u and v in G , by d G ( u ) the degree of vertex u . The Hosoya polynomial of G is H ( G ) = ∑ { u , v } ⊆ V ( G ) x d G ( u , v ) . For any positive numbers m and n , the partial Hosoya polynomials of G are H m ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u ) = d G ( v ) = m x d G ( u , v ) , H m n ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u ) = m , d G ( v ) = n x d G ( u , v ) . It has been shown that H ( G 1 ) − H ( G 2 ) = x 2 ( x + 1 ) 2 ( H 3 ( G 1 ) − H 3 ( G 2 ) ) , H 2 ( G 1 ) − H 2 ( G 2 ) = ( x 2 + x − 1 ) 2 ( H 3 ( G 1 ) − H 3 ( G 2 ) ) and H 23 ( G 1 ) − H 23 ( G 2 ) = 2 ( x 2 + x − 1 ) ( H 3 ( G 1 ) − H 3 ( G 2 ) ) 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 ( 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].