The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. Denote by  and  the Hosoya index and the Merrifield-Simmons index of  a graph G. Then k-tree-triangle graph  is defined as a connected graph with ktriangles such that each pair of triangles has at most one common vertex. In this paper, the maximal and minimal Hosoya index and Merrifield-Simmons index for a graph  are determined. Denote by  the set of all tree-triangle graphs with ktriangles and diameter  Among all graphs in  we characterize the graphs with the maximal Merrifield-Simmons index and the graphs with the minimal Hosoya index. In the end, we give an algorithm to count the number of the Hosoya index and Merrifield-Simmons index of

Hosoya index, Merrifield-Simmons index, tree-triangle graph, diameter, algorithm.