The number of independent sets of unicyclic graphs with given matching number

The Hosoya index z(G)z(G) of a graph GG is defined as the number of matchings of GG and the Merrifield–Simmons index i(G)i(G) of GG is defined as the number of independent sets of GG. Let U(n,m)U(n,m) be the set of all unicyclic graphs on nn vertices with α′(G)=mα′(G)=m. Denote by U1(n,m)U1(n,m) the graph on nn vertices obtained from C3C3 by attaching n−2m+1n−2m+1 pendant edges and m−2m−2 paths of length 2 at one vertex of C3C3. Let U2(n,m)U2(n,m) denote the nn-vertex graph obtained from C3C3 by attaching n−2m+1n−2m+1 pendant edges and m−3m−3 paths of length 2 at one vertex of C3C3, and one pendant edge at each of the other two vertices of C3C3. In this paper, we show that U1(n,m)U1(n,m) and U2(n,m)U2(n,m) have minimal, second minimal Hosoya index, and maximal, second maximal Merrifield–Simmons index among all graphs in U(n,m)∖{Cn}U(n,m)∖{Cn}, respectively.