Merrifield-simmons index of a class of unicyclic graphs

Abstract

It is well known that the graph invariant, the Merrifield-Simmons index is important in structural chemistry. 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. We characterize the n-vertex unicyclic graphs whose vertices on its unique cycle all have degree at least three with the first, the second and the third largest Merrifield-Simmons indices, respectively.