Study On The Bounds Of Several Kinds Of Energy Of Graphs

In this paper,we investigate the bounds of several kinds of energy of the graphs,such as,the upper bound of the Laplacian energy and the Laplacian-energy-like invariant of the DoubleStar graphs,the bound of the signless Laplacian energy of the graph,the(sharp)lower bounds of the distance energy,the distance Laplacian energy,and the distance signless Laplacian energy.In Chapter 1,we introduce the research background,progress significance,and the current research situation of graph theory at home and abroad.In Chapter 2,firstly,the Laplacian characteristic polynomials of graphs when the number of leaves in left and right star graphs is equal and unequal are discussed.And then,by using the solution theory of algebraic equation,the Laplacian energy and the exact expressions of Laplacian-energy-like invariant and Laplacian Estrada index of the Double-Star graphs are obtained whe n the number of leaves on both sides is equal.Furthermore,the upper bounds of Laplacian energy and Laplacian-energy-like invariant are given when the number of leaves on both sides is unequal.In Chapter 3,the upper and lower bounds of the signless Laplacian energy of the graph are studied.When the matrix which consisted of the signless Laplacian matrix subtract the average degree diagonal matrix is non-singular,combined with algebraic operation and monotonicity of functions,the upper and lower bounds of the signless Laplacian energy of the graph are obtainned,and we illustrate that the proposed results are better than some early conclusions with an example.In Chapter 4,we get the distance energy,the distance(signless)Laplacian energy of the graph,as well as the structure of the graph when equality holds by using the algebraic operation,monotonicity of functions.Finally,some examples are given to illustrate that the results of this paper are better than some existing conclusions.