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

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