Bipartite Graph To The Study Of The Zagreb Index

Nowadays, combinatorial chemistry is a new powerful technology in the fields of drug design and molecular systemization. It aims at discovery of compounds those have a certain chemical and biological activity. Topological indices can reflex chemical and physical properties of molecules statistically. Since different topological indices have different structural interpretations, many kinds of topological indices are introduced in chemistry. So the study of topological indices of molecules have important theoretical value and application background. This thesis mainly study topological indices of bipartite graphs.In Chapter 1, after introducing some topological indices and basic terminology and notations, we give a brief overview to the main results of the thesis.In Chapter 2, we characterize the bipartite graphs with given number of vertices and edges which have the smallest and largest values of the sum of squares of degrees. Meanwhile, we present some bipartite graphs attaining extreme values.In Chapter 3, we characterize the bipartite graphs with given number of vertices and edges which have the smallest values of the first general Zagreb index, and give some necessary conditions for the first general Zagreb index of bipartite graphs attaining the largest values.In Chapter 4, we give some necessary conditions for simple graphs to have the smallest values of the second Zagreb index. Meanwhile, we give the smallest values of the second Zagreb index of bipartite graphs with n vertices and m (m < n) edges. Some corresponding extremal graphs are also presented.We conclude the thesis with Chapter 5 by proposing some further research problems.