The Hosoya index of a graph is defined as the total number of the matching of the graph. In this paper, the ordering of polygonal chains with respect to Hosoya index is characterized.
For a graph G, let h(G;x) = h(G) and [G]h denote the adjoint polynomial and the adjoint equivalence class of G, respectively. In this paper, a new application of [G]h is given. Making use of [G]h, we give a necessary and sufficient condition for adjoint uniqueness of the graph H such that H = G, where H = ( i∈A Pi) ( j∈B Uj), A ■ A = {1,2,3,5} {2n|n ∈ N,n ≥ 3}, B ■ B = {7,2n|n ∈ N,n ≥ 5} and G = aP1 a0P2 a1P3 a2P5 ( in=3aiP2i).
