Ordering Of Energies For Signed Graphs With Few Cycles

The study of graph theory began in the 1950 s.After only a few decades of development,graph theory has become a systematic and comprehensive theoretical branch.The energy of a graph is an important parameter related to the spectrum in the graph,which is defined as the sum of the absolute values of all the eigenvalues of the graph.The energy of the graph comes from the research field of chemistry,and chemists have found that the energy of most hydrocarbon molecules is linearly related to the total electric energy of this molecule.Because graph energy has a significant physical and chemical background,it has attracted the attention of many chemical workers and mathematical workers,has become a hot topic.The minimal energy of bicyclic bipartite signed graphs、bicyclic signed graphs and tricyclic signed graphs is studied by the inversion of graph theory and the upper and lower bound of energy.This thesis is divided into four chapters.The first chapter is the introduction,which briefly introduces the concepts and basic knowledge of graph,as well as the main research conclusions of the few circle graphs,and gives the main problems to be discussed in this paper.In the second chapter,we mainly study the first four small energy of bipartite bicyclic signed graphs with 9)vertices and the third and fourth minimal energy.In this paper,we divide the bicyclic graphs into three types.Firstly,the bipartite bicyclic signed graphs determines the first four minimal energies by comparing the adjacency matrix with the coefficient inversion methods.Then,the lower bound of the signed graph in the corresponding case is obtained by means of the eigenvalue.In the third chapter,we summarize the main research results of this paper.Firstly,the tricyclic graphs are divided into three cases.and then the description of the tricyclic graphs with minimal energy is obtained by using the energy comparison method of the symbolic graph.Finally,the conjecture of minimal energy of tricyclic graphs are solved.In the fourth chapter,we summarize the prospect of solving the minimal energy methods.The ideas of solving the minimal energy of 4 cycles graphs is presented.