The Hosoya index z(G) of a graph G is defined as the number of matchings of G and the MerrifieldSimmons index i(G) of G is defined as the number of independent sets of G. Let U(n,m) be the set of all unicyclic graphs on n vertices with α′(G)=m. Denote by ~(U1)(n,m) the graph on n vertices obtained from~(C3) by attaching n-2m+1 pendant edges and m-2 paths of length 2 at one vertex of~(C3). Let ~(U2)(n,m) denote the n-vertex graph obtained from~(C3) by attaching n-2m+1 pendant edges and m-3 paths of length 2 at one vertex of ~(C3), and one pendant edge at each of the other two vertices of ~(C3). In this paper, we show that ~(U1)(n,m) and ~(U2)(n,m) have minimal, second minimal Hosoya index, and maximal, second maximal MerrifieldSimmons index among all graphs in U(n,m)~(Cn), respectively.
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