On The Research Of The Energy Of Graphs

Let G be a simple undirected graph.The energy of graph G is defined as the sum of the absolute values of all eigenvalues of the graph.The moving-star operation is a kind of star moving between two vertices of a graph.The moving-path of length 2 operation is a kind of path of length 2 moving between two vertices of a graph.In Chapter 2,the subgraph grafting operation on a graph is a kind of subgraph grafting moving between two vertices of a graph.In this paper,we study how the bipartite graph energy and quasi-order changes under the above graph grafting.As the application of these operations,we study the orders of energies in the two classes of trees.In Chapter 1,we introduce the development of the research of spectral theory and energy of graph,and then we list some basic definitions,symbols and notations which will be used in this thesis.In Section 3,we introduce the present situation of the study on the graph energy.In Section 4,we introduce the main results of this paper.In Chapter 2,we study the effect of moving-path of length 2 operation in bipartite graph.In Section 2,we give some formulas for characteristic polynomials of the coalescence of some bipartite graphs.Then we apply the formulas to study the effect of subgraph moving in bipartite graph.In Chapter 3,we study a new method to directly compare the energies of two bipartite graphs which does not use the quasi-order of graphs.It provides an important tool for Chapter 4.In Chapter 4 and Chapter 5,we study the changes of domination number of the coalescence of two graphs.Then we apply the wounded spider operation and the above results to study the order of the trees with given order and domination number.We determine the trees with the third and fourth minimal energies in the set of trees with given order and domination number.And also determine the minimal energies of the hand-in-hand spider graph.