On The Hosoya Index And The Merrifield-Simmons Index Of Tree-Triangle Graphs And K₄-Tree Graphs

The molecular topology has formed a set of strict theory system. In recent years, more and more mathematicians and chemists use the knowledge of graph theory to solve the problem of molecular topological index. Many topology indices have been studied directly. Among these indices, the Hosoya index and the Merri?eld-Simmons index are important two indices, which are used to study the total number of matches of molecular graphs and independent sets, respectively.Both the Hosoya index and the Merri?eld-Simmons index have close relationships with the physical and chemical properties, such as the total π-electronic energy of molecular and the boiling point. Therefore, many mathematicians and chemists have began to study them. And ordering problem and characterizing of the extremal graphs are an important research issue, especially for the tree. Let Gk be a connected graph with k triangles such that each pair of triangles has at most one common vertex. The graph Gkis called Tree-triangle. Let T be a connected graph with K4 such that each pair of K4 has at most one common vertex.The graph T is named K4-tree. In the thesis, we mainly study that Gkand T compare with the known results about the Hosoya index and the Merri?eldSimmons index of tree. And we suppose whether it can be extended to the tree of other completely-tree graphs.The thesis consists of ?ve chapters: Chapter one mainly introduces the definitions of tree-triangle graph and K4-Tree graph, and the backgrounds of the Hosoya index and the Merri?eld-Simmons index. And brie?y introduces the purposes to study the tree-triangle graph and K4-Tree graph. Chapter two gives the algorithms about the Hosoya index and the Merri?eld-Simmons index for treetriangle graphs, and give the applications of them. Chapter three illustrates the bounds of these two indices for the tree-triangle graph, and we determine the graphs with the minimum Hosoya index and the maximum Merri?eld-Simmons index among the set Gkwith ?xed diameter, respectively. At the same time, we calculate the recurrence relations of the two indices for the tree-triangle graphs by the related formulas. Chapter four gives the algorithms about the Hosoya index and the Merri?eld-Simmons index for the K4-Tree graph, and give two examples.Chapter ?ve presents the conclusion and the prospect of further research.