The extremal fully loaded graphs with respect to Merrifield-Simmons index

The Merrifield-Simmons index, denoted by i(G), of a graph G is defined as the total number of its independent sets of G. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than 3 in its unique cycle. Let u(n)(1),be the set of fully loaded unicyclic graphs. In this paper, we determine graphs with the largest, second-largest, and third-largest Merrifield-Simmons index in u(n)(1).