Automorphisms Of Several Algebraic Graphs On Matrix Algebra

Algebraic graph combines algebra with graph theory,prompting the development of both subjects.Matrix theory,group theory,etc.,promote and deepen the investigation of combinatorial properties of graphs;By constructing graphs on algebraic structures,such as zero-divisor graph,commuting graph,total graph and so on,the properties of these graphs can solve some problems which is difficult to solve in algebra theory.Automorphisms of graphs explore the structures of graphs,especially the symmetry of graphs.It is helpful to explore the structures of graphs by investigating the automorphism group of graphs applying the knowledge of algebra.As we know the preserving problem is a important and deeply investigated subject in algebra.And it converts the preserving problem in algebraic systems into characterizing the graph automorphisms by constructing some graphs on algebraic systems,such as ring or group and so on.So it is significant to investigate the automorphism group of some graphs on algebraic systems.Especially,it is both important and feasible to investigate algebraic graphs on matrix algebra,which has preferable structures and rich properties.This dissertation investigates the automorphisms of some algebraic graphs including following five chapters.Chapter 1 introduces the background and the significance of the subject chosen.Besides,the main results of this dissertation,the main methods and notations used in this dissertation are also explained.Chapter 2 investigates the automorphisms of two kinds of algebraic graphs,one of which is the zero-divisor graph on the algebra Nn(Fq)consisting of all strictly upper triangular matrices of order n over a finite field Fq,denoted by ?(Nn(Fq));the other is zero-divisor graph on the algebra Tn(Fq)consisting of all upper triangular matrices of order n over a finite field Fqbased on the ideal I = {aE1n},denoted by ?I(Nn(Fq)).It is proved that for almost all n any automorphism of these two kinds of graphs can be written as the compositions of three standard automorphisms: inner automorphisms,field automorphisms and singular automorphisms.Chapter 3 investigates the automorphisms of two kinds of algebraic graphs,of which one is the inclusion ideal graph on the algebra Mn(Fq)consisting of full matrices of order n over a finite field Fq,denoted by Iin(Mn(Fq));the other is the ideal-relation graph on the block upper triangular algebra Br(Fq)over a finite field Fq,denoted by Ire(Br(Fq)).It is proved that for n 3 any automorphism of Iin(Mn(Fq))can be written as the compositions of two standard automorphisms: ideal right regular automorphisms and ideal field automorphisms;And for r 3 any automorphism of Ire(Br(Fq))can be written as the compositions of two standard automorphisms: reversal automorphisms and singular automorphisms.Chapter 4 investigates the automorphisms of co-maximal ideal graph on the algebra of Mn(Fq),denoted by C(Mn(Fq)).It is proved that for n 3 any automorphism of C(Mn(Fq))can be written as the compositions of two standard automorphisms: ideal right regular automorphisms and ideal field automorphisms.The last chapter,Chapter 5,summarizes the main conclusions of this dissertation and give a outlook for the subject investigated in this dissertation.