Cacti with the maximum Merrifield–Simmons index and given number of cut edges

The Merrifield–Simmons index of a graph GG, denoted by i(G)i(G), is defined to be the total number of its independent sets, including the empty set. A graph GG is called a cactus if each block of GG is either an edge or a cycle. Denote by C(n,k)C(n,k) the set of connected cacti possessing nn vertices and kk cut edges. In this work, we shall characterize the cacti with the maximum Merrifield–Simmons index among all graphs in C(n,k)C(n,k) for all possible values of kk.