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

• Authors: Gong Chen;Zhongxun Zhu
• Venue: Discrete Applied Mathematics
• Year: 2012

The Hosoya index z(G) of a graph G is defined as the number of matchings of G and the Merrifield-Simmons 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 @a^'(G)=m. Denote by U^1(n,m) the graph on n vertices obtained from C"3 by attaching n-2m+1 pendant edges and m-2 paths of length 2 at one vertex of C"3. Let U^2(n,m) denote the n-vertex graph obtained from C"3 by attaching n-2m+1 pendant edges and m-3 paths of length 2 at one vertex of C"3, and one pendant edge at each of the other two vertices of C"3. In this paper, we show that U^1(n,m) and U^2(n,m) have minimal, second minimal Hosoya index, and maximal, second maximal Merrifield-Simmons index among all graphs in U(n,m)@?{C"n}, respectively.