On the extremal Merrifield–Simmons index and Hosoya index of quasi-tree graphs

It is well known that the two graph invariants, “the Merrifield–Simmons index” and “the Hosoya index” are important in structural chemistry. A graph GG is called a quasi-tree graph, if there exists u0u0 in V(G)V(G) such that G−u0G−u0 is a tree. In this paper, at first we characterize the nn-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Merrifield–Simmons indices. Then we characterize the nn-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Hosoya indices, as well as those nn-vertex quasi-tree graphs with kk pendent vertices having the smallest Hosoya index.