The Merrifield-Simmons indexi(G)of a graphGis defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets ofGThe Hosoya indexz(G)of a graphGis defined as the total number of independent edge subsets, that is, the total number of its matchings. ByC(n,k,λ)we denote the set of graphs withnvertices,kcycles, the length of every cycle isλ, and all the edges not on the cycles are pendant edges which are attached to the same vertex. In this paper, we investigate the Merrifield-Simmons indexi(G)and the Hosoya indexz(G)for a graphGinC(n,k,λ).