Automorphisms And Domination Numbers Of Several Classes Of Algebraic Graphs

Frucht proposed that any group can be regarded as an automorphism group of a graph in 1938.This research transformed the study of groups into the automorphism groups of relative graphs,which also became the beginning of studying the automorphism groups of algebraic graphs.It is known that an algebraic system usually have a good structure,thus the algebraic graphs associated with algebraic systems will also maintain good symmetry,which allows us to obtain better properties of algebraic graphs when studying the automorphism groups of them.Therefore,the study of automorphism groups is of great significance both in algebra and graph theory.The domination theory of graphs is widely used in computers,coding theory and communication networks.The dominating set and the domination number of algebraic graphs are also particularly important in algebraic graph theory.There are five chapters in this thesis,which investigate the automorphisms and domination numbers of some algebraic graphs:In Chapter 1,the background and the significance of the thesis are introduced.In addition,the main results of this thesis,the main methods and the basic definitions and related notations are presented as well.In Chapter 2,some properties of the subspace inclusion graph of a vector space In(V),defined by Das,are investigated:(1)A lower bound of the domination number of the graph In(V)are given;(2)The theorem that the graph In(V)is a distance regular graph when the dimension of the vector space V is 3 is proved;(3)The automorphism of the graph In(V)are characterized.In Chapter 3,the transformation graph Γt(V)over vector space V are defined,and some investigations are done:(1)The domination number of the graph Γt(V)are given;(2)The automorphisms of the graph Γt(V)are characterized.In Chapter 4,the one-devisor graph Tn(q)over the semigroup of n × n upper triangular matrices are defined,and the automorphism group of the graph Tn(q)are characterized.In Chapter 5,we summarize the main conclusions of this paper and introduce the relevant contents for subsequent research.