The Harary Indices Of Unicyclic Graphs With Given Matching Number And Trees With Described Diameter

In the1960s or1970s, as a branch of mathematics, the graph theory has experienced the explosion growth. It has extensive applied in chemistry, physics, biology, network design, information science, computer science and other fields. Research on the molecular topological index problem is an important part of graph theory application fields. The topological index is a numerical method of molecular structure, and a graph invariants. It directly generated from the molecular structure, reflecting the structural features of the compounds. We can use mathematical methods to study the topological properties of molecular structure and deal with some of our life and production related problems in Chemistry. We principally focus on the Harary index of graph in this thesis, which is one of the topological index in graph theory and has many applications in various areas.In this paper, we mainly discuss the Harary indices of unicyclic graphs with given matching number and trees with described diameter.The first chapter, mainly introduce the fields of application of topological index and its significance, reviewing some investigative results about Harary index in recent.years.The second chapter, mainly discuss a series of transformations of unicyclic graph with given matching number, and get the extremal Harary index among the unicyclic graph with given matching number.The third chapter, mainly study some transformations lemmas of trees with described diameter, and obtain the extremal Harary index among the trees with described diameter.