On the Hosoya index of graphs

Let G be a (molecular) graph. The Hosoya index Z(G) of G is defined as the number of subsets of the edge set E(G) in which no two edges are adjacent in G,i.e.,Z(G) is the total number of matchings of G. In this paper,we determine all the connected graphs G with n +1 ≤ Z(G) < 5n - 17 for n ≥ 19. As a byproduct,the graphs of n vertices with Hosoya index from the second smallest value to the twenty first smallest value are obtained for n ≥ 19.