Graphs Whose Energy Does Not Exceed 6

In Graph Theory,the research of energy is very active,which has been developed fast and attracted much attention in recent decades.The definition of graph energy was first proposed by Gutman in 1978.Let G be a simple graph,then the definition of the energy of G is ???,Where?₁,…,?_nare the eigenvalues of G.There exist many connections between the graph energy and some graph parameters,such as the rank,matching number,chromatic number and so on.In this paper,based on the characterization of graphs with rank 4 due to Chang et al.[27],we mainly characterize the connected graphs with energy less than 6 and the graphs whose energy are precisely 6.The main contents of the thesis are as follows:The first chapter mainly introduces the background and significance of the research,the research status and the basic concept of the graph.In chapter 2,we mainly studies the conclusion and proof of the lower bound of graph energy.In chapter 3,we give the main theorem and proof of this paper,i.e.,characterize the graphs with energy less than 6 and the graphs whose energy are precisely 6.The fourth chapter summarizes the main contents of this thesis and puts forward the further research objectives in the future.