For a graph , the Hosoya index and the Merrifield–Simmons index are defined as the total number of its matchings and the total number of its independent sets, respectively. In this paper, we characterize the structure of those graphs that minimize the Merrifield–Simmons index and those that maximize the Hosoya index in two classes of simple connected graphs with vertices: graphs with fixed matching number and graphs with fixed connectivity.
This project was supported by the Special Fund for Basic Scientific Research of Central Colleges, South-Central University for Nationalities (CZZ13006). The fourth author was supported by the National Research Foundation of South Africa, grant number 70560.