矩阵代数上几类代数图的自同构

发布时间：2018-04-15 18:39

本文选题：图自同构 + 零因子图　；参考：《中国矿业大学》2017年博士论文

【摘要】：代数图论将代数和图论结合起来,促进了两个学科的共同发展.代数中矩阵理论,群论等理论促进加深了对图的组合性质的研究;在代数结构上构造各类图,如零因子图、交换图、全图等,这些代数图的性质也可以解决用代数理论不易解决的代数问题.图的自同构揭示图的结构,特别是图的对称性,因此利用代数理论研究图的自同构群对揭示图的结构有重要的意义.众所周知,保持问题是代数中一个有重要意义且研究深入的问题.而在一些代数结构(比如环,群等)上定义相关的图,对该图的自同构群的研究就是代数中的保持问题.因此研究代数结构上的各类图的自同构问题有着重要的意义.尤其是矩阵代数,由于它具有较好结构和丰富的性质,研究其上的代数图的自同构兼具重要性和可行性.本篇学位论文研究了矩阵代数上几类图的自同构刻画问题,共分为五章.具体的研究内容介绍如下.第1章是绪论部分,介绍了本论文的选题意义及研究背景,论文的主要工作,主要的研究方法以及本论文中的符号约定.第2章研究两类代数图的自同构,一是有限域Fq上的由n阶严格上三角矩阵全体构成的代数Nn(Fq)上的零因子图Γ(Nn(Fq))的自同构;二是有限域Fq上由n阶上三角矩阵全体构成的代数Tn(Fq)上的基于理想I={aE1n}的零因子图ΓI(Tn(Fq))的自同构.证明了除阶数n较小的个别情况外,这两类图的任意自同构均可以用三种标准自同构:内自同构,域自同构和奇异自同构的复合表示.第3章研究另两类代数图的自同构,一是有限域Fq上的由n阶全矩阵全体构成的代数Mn(Fq)上的理想包含图Iin(Mn(Fq))的自同构;二是有限域Fq上的分块上三角代数Br(Fq)上的理想关系图Ire(Br(Fq))的自同构.证明了当n 3时,图Iin(Mn(Fq))的任意自同构均可以用其上的两种标准自同构:理想右正则自同构和理想域自同构的复合表示;当r 3时,图Ire(Br(Fq))的任意自同构均可以用其上的两种标准自同构:翻转自同构和奇异自同构的复合表示.第4章研究全矩阵代数Mn(Fq)上的理想互极大图C(Mn(Fq))的自同构.证明了当n 3时,图C(Mn(Fq))的任意自同构均可以用其上的两种标准自同构:理想右正则自同构和理想域自同构的复合表示.第5章对本论文的主要结论进行了总结,并对本论文的研究课题的前景进行了展望.
[Abstract]:Algebraic graph theory combines algebra and graph theory and promotes the common development of the two disciplines.The theory of matrix and group theory in algebra promotes the study of the combinatorial properties of graphs, constructs all kinds of graphs on algebraic structure, such as zero digraph, commutative graph, total graph, etc.The properties of these algebraic graphs can also solve algebraic problems which are difficult to solve by algebraic theory.The automorphism of a graph reveals the structure of a graph, especially the symmetry of a graph, so it is important to study the automorphism group of a graph by using algebraic theory to reveal the structure of a graph.As we all know, the problem of preserving is an important problem in algebra.Some related graphs are defined on some algebraic structures (such as rings, groups, etc.). The study of the automorphism group of the graph is the problem of preserving in algebra.Therefore, it is of great significance to study the automorphism of various graphs on algebraic structures.Especially for matrix algebra, because of its good structure and rich properties, it is important and feasible to study the automorphism of algebraic graph on matrix algebra.In this dissertation, we study the automorphism characterizations of some kinds of graphs on matrix algebra, which are divided into five chapters.The specific research content is as follows.The first chapter is the introduction, which introduces the significance and research background of this thesis, the main work, the main research methods and the symbolic agreement in this paper.In chapter 2, we study the automorphism of two kinds of algebraic graphs. One is the automorphism of Zero-Different graphs 螕 _ (NnNn _ (FQ)) over a finite field F _ Q, which consists of all algebraic upper triangular matrices of order n ~ (n).The second is the automorphism of the Zero-Divisor Graph 螕 _ I _ (T _ (n) F _ Q) based on the ideal I = {aE1n} over the algebra T _ n _ F _ Q over the finite field F _ Q consisting of all the upper triangular matrices of order n.It is proved that the arbitrary automorphism of these two kinds of graphs can be represented by three kinds of standard automorphisms: inner automorphism, domain automorphism and singular automorphism, except in some cases where the order n is small.In chapter 3, we study the automorphism of the other two kinds of algebraic graphs. One is the automorphism of the ideal inclusion graph I _ (n) MnMN _ (F _ Q) on the finite field F _ Q, which is composed of all the n-order total matrices.The second is the automorphism of the ideal relation graph Iren Br-Fqg over the block upper triangular algebra Br-FQ over the finite field FQ.It is proved that if n 3, any automorphism of a graph I _ (n _ n _ n _ n) can be represented by two kinds of standard automorphisms: ideal right regular automorphism and ideal domain automorphism, and when r _ 3, the arbitrary automorphism can be represented by two kinds of standard automorphisms: ideal right regular automorphism and ideal domain automorphism.The arbitrary automorphism of a graph Irechon (Fqn) can be represented by two kinds of standard automorphisms on it: reversed automorphism and singular automorphism.In chapter 4, we study the automorphism of the ideal mutual maximal graph C _ (n) mn _ (F _ (Q)) over the total matrix algebra Mn-F _ (Q).It is proved that any automorphism of graph C _ (mn) F _ (Q) can be represented by two kinds of standard automorphisms on n _ 3: ideal right regular automorphism and ideal domain automorphism.In chapter 5, the main conclusions of this paper are summarized, and the prospect of this thesis is prospected.
【学位授予单位】：中国矿业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O151.21;O157.5

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