Study Of Generalized Distance Energy Of Graphs And Energy Of Chain Graph

Graph energy is graphs based spectral invariant.It is an important research branch in the field of graph theory and application,which can expand the research content of graph theory.Since Gutman proposed the adjacency matrix energy of undirected graph in 1977,many scholars have proposed various energies,such as distance energy,Laplacian energy and generalized distance energy.By studying the bounds of various energies and depicting extreme value graphs,we can reduce the experimental links in chemical research and determine the stability of compounds.Based on the existing research,this paper expands the parameter range and simplifies the parameters in the bound,and obtains the upper and lower bounds of the generalized distance energy of the graph and the energy of the chain graph.The main work is as follows:In Chapter 1,we introduce the research significance and background of graph energy,the related basic concepts of graph,and expounds the research status of graph generalized distance energy,chain graph energy and Laplacian energy at home and abroad;In Chapter 2,firstly,using the properties of convex linear combination,we study how parameters affect the upper and lower bounds of generalized distance energy.Then,the main characteristics of distance energy are studied.Secondly,By analyzing the relationship between distance energy,unsigned distance,Laplacian energy and generalized distance energy,the upper and lower bounds are expressed by simple parameters such as vertex,number of edges,diameter and determinant,and the extreme graphs is characterized.Finally,by combining the properties of unsigned Laplacian spectrum and generalized distance spectrum,the generalized distance energy of complete k-part graph and Petersen graph is obtained;In Chapter 3,we analyze the mathematical properties of chain graph,studies the particularity of chain graph adjacency matrix and Laplacian matrix,and optimizes the upper and lower bounds of chain graph energy and Laplacian energy.Then bipartite chain graphs are classified according to their common characteristics,so that the Laplacian energy can be expressed by algebraic connectivity(sub small Laplacian eigenvalue),and then the maximum Laplacian energy of all connected bipartite chain graphs can be obtained by comparing algebraic connectivity.