Further results on the Merrifield–Simmons index

Abstract

The total number of independent subsets, including the empty set, of a graph, is also termed as the Merrifield–Simmons index (MSI) in mathematical chemistry. The connective eccentricity index (CEI), the multiplicative Wiener index (MWI), and the Wiener polarity index (WPI) are all distance-based topological indices. Some of these topological indices found applications in chemistry. The eccentric complexity of a graph G is defined to be the number of different eccentricities of its vertices. In this paper, we first investigate the relationship between MSI and three other distance-based topological indices (CEI, MWI, WPI). We prove that MSI>WPI for any tree, MSI>CEI for any connected bipartite graph, and MWI>MSI for connected graphs with given restricted condition on size. Then we consider extremal problems of MSI on graphs with small eccentric complexity. Moreover, we prove an inequality relating MSI, independence number and eccentric complexity for some special graph families. Finally, some further problems and conjectures are proposed.