The Hosoya Index Of A Graph With K Cut Edges

The Hosoya index of a graph is defined to be the total number of its matchings(or independent edge subsets).The Hosoya index is a very important molecular topological. It has a close relationship with the physical and chemical characteristics of compound which is represented by molecular, for instance, the totalπ-electron energy, the boiling points, etc.The bound of the Hosoya index of some specific graphs is one of the important task about the study of the graph theory.This thesis mainly discusses the lower bound of the Hosoya index of a class of tree, which has two vertices with maximum degree; unicyclic graphs, bicyclic graphs and connected graphs with given cut edge.The structure of this thesis as follows:In Chapter 1, we introduce a background of the Hosoya indices, some useful concepts and terminology, the questions and correspond results in this thesis.In Chapter 2, we discuss the lower bound of the Hosoya index of unicyclic and bicyclic graphs with given number of cut edges.In section 2.1, we introduce the lower bound for the Hosoya index of a class of unicyclic graphs and the correspond extreme graph; in section 2.3 and 2.4,we respectively get the lower bound for the Hosoya index of the unicyclic and bicyclic graphs with given cut edges, then give the correspond extreme graphs.In Chapter 3, we discuss the lower bound for the Hosoya index of graphs with k cut edges.In section 3.1 and 3.2,we study the sharp lower bound for the Hosoya index of 2-edge-connected graph and correspond extreme graphs.In Chapter 4, we discuss the Hosoya index of a class of tree, which has two vertices with maximum degree, and determine the correspond extreme graph.