Graphs with given number of cut vertices and extremal Merrifield–Simmons index

The Merrifield–Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Denote by G(n,k)G(n,k) the set of connected graphs with nn vertices and kk cut vertices. In this paper, we characterize the graphs with the maximum and minimum Merrifield–Simmons index, respectively, among all graphs in G(n,k)G(n,k) for all possible kk values.