| Determining whether a graph possess certain structural properties is a significant problem in Graph Theory.Frankly speaking,researches on the Hamiltoniacity,Hamilton-connected,matching extension and issues with respect to indepen-dent number have become the challenging problems in this field.Topological indices are graph invariants which are closely related to the structural properties of the corresponding graphs.The forgotten index of a graph is one of the most important topological indices,which is the sum of the cubes of all vertex degrees in the corresponding graph:F(G)=(?)(d_G(u))~3.In this paper,we mainly explore sufficient conditions for graphs to have some special structural properties in terms of the forgotten index of graphs.In Chapter 1,we first introduce the development of Graph Theory,and then propose the basical concepts and terminologies.Several topological indices and its applications are also presented.In Chapter 2,we attempt to find sufficient conditions for general graphs to be k-edge-hamiltonian and k-path-coverable respectively in terms of the forgotten index,the corresponding extremal graphs are also characterized.In Chapter 3,we explore sufficient conditions for general graphs to be traceable and Hamilton-connected by using of the forgotten index,and present the corresponding extremal graphs.In Chapter 4,we propose a new topological invariant of graphs-the deficiency range of matchings,and determine the values of the deficiency range of matchings for several special graph classes.In the last part of this thesis,we summarize the main results obtained and propose several problems which will be considered in the future. |
