For a simple graph G without K3 subgraph, by means of discussing a method of 1-1 mapping and a method of complete components, theauthor proves that its k-matching number φk with its the number N(G,n−k) of S(n)-factors with exactly (n − k) components is equal, and its the Hosoyaindex Z(G) with its the number A(G) of S(n)-factors is equal through analyzing the relations k-matching of a simple graph G without K3 subgraph withexactly (n−k) components; Finally, the author solves the recurrence relation of the number φk,Tt of k-matching of the regular m-furcating tree, so getsits the matching defect polynomials m(T,x); and a recurrence formula of the Hosoya index Zt of the regular m-furcating tree; and a counting formula ofthe Hosoya index ZG of the graph by q chord graph of cycle. Simultaneously the corresponding examples are given.