Abstract

It is well known that the two graph invariants, “the Merrifield–Simmons index” and “the Hosoya index” are important in structural chemistry. A graph G is called a quasi-tree graph, if there exists u 0 in V ( G ) such that G − u 0 is a tree. In this paper, at first we characterize the n -vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Merrifield–Simmons indices. Then we characterize the n -vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Hosoya indices, as well as those n -vertex quasi-tree graphs with k pendent vertices having the smallest Hosoya index.

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