On the Hosoya index and the Merrifield–Simmons index of graphs with a given clique number

The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets (including the empty vertex set) of the graph, respectively. Let Wn,kWn,k be the set of connected graphs with nn vertices and clique number kk. In this note we characterize the graphs from Wn,kWn,k with extremal (maximal and minimal) Hosoya indices and the ones with extremal (maximal and minimal) Merrifield–Simmons indices, respectively.