几类代数图的自同构问题的研究

发布时间：2018-01-08 23:00

  本文关键词：几类代数图的自同构问题的研究　出处：《中国矿业大学》2016年博士论文　论文类型：学位论文

  更多相关文章： 零因子图 广义Cayley图 交换图 全图 自同构

【摘要】：代数图论是一门应用代数方法来解决图论问题的学科,它促进了代数学和图论两个学科的发展.在研究代数学中的非线性保持问题时,由于所研究的对象为非线性映射,故研究难度很大,而研究代数图的自同构问题有助于研究对应代数系统上的非线性保持问题,且会较容易表述并刻画出非线性映射.所以,本论文研究代数图的自同构问题有着重要意义.本文主要研究了几类代数图的自同构问题及n阶全矩阵代数Mn(R)的由对合矩阵决定的线性映射,共分为7章.具体研究内容按照章节介绍如下第1章是绪论部分,介绍了本论文的选题意义,论文主要工作,主要研究方法和符号约定.第2章研究了矩阵环上的零因子图的自同构问题,其中2.3节研究了n阶上三角矩阵环Tn(Fq)的零因子图的自同构,改进了文献[3]的Tn(Fq)的环边零因子图的自同构结论；2.4节研究了二阶矩阵环M2(Fq)的零因子图的自同构,纠正了[4,定理3.9]和[5,定理3.8]的核心错误；2.5节研究了n阶矩阵环Mn(Fq)的零因子图的自同构,其中n≥3.第3章研究了上三角矩阵半群Tn(q)的广义Cayley图GCay(Tn(q))的自同构群.构造了GCay(Tn(q))的两种自同构σJ和τ,之后证明GCay(Tn(q))的任意自同构均可由这两个自同构表示出来,并给出具体公式.第4章研究了二阶矩阵环的交换图Γ(M)的自同构群.首先,基于Γ(M)我们关联一个压缩图ΓE(M),然后研究Aut(Γ(M))和Aut(ΓE(M))之间的关系并求出Aut(ΓE(M)),最后刻画Γ(M)的自同构群.第5章研究了二阶矩阵环M2(Fq)的全图T(Γ(M2(Fq)))的自同构.当char(Fq)≠ 2时,构造了T(Γ(M2(Fq)))的四种自同构—τ,LP,RP,f,然后证明T(Γ(M2(Fq)))的任意自同构均可分解成这四种自同构的乘积,并给出具体表达式.第6章弱化了n阶全矩阵代数Mn(R)的由单位积决定的线性映射(参见文献[6])的限制条件,研究了Mn(R)由对合矩阵决定的线性映射.将文献[6]研究的Mn(R)的保单位积线性映射ψ和在单位阵处可导的线性映射φ分别弱化为保对合矩阵的线性映射ψ和对合矩阵处可导(也即：对合矩阵处Jordan可导)的线性映射φ,推广了文献的结论.第7章详细总结了本学位论文的核心结论.
[Abstract]:Algebraic graph theory is a subject that applies algebraic methods to solve graph theory problems. It promotes the development of algebra and graph theory. Because the object studied is nonlinear mapping, it is very difficult to study the automorphism of algebraic graph. And it is easier to describe and depict the nonlinear mapping. In this paper, it is of great significance to study the automorphism of algebraic graphs. In this paper, we mainly study the automorphism of several algebraic graphs and the linear mappings determined by involutive matrices of all matrix algebras of order n. It is divided into seven chapters. The specific research contents are introduced as follows: chapter 1 is the introduction part, which introduces the significance of the topic of this paper, the main work of the paper. Chapter 2 deals with the automorphism of zero factor graphs over matrix rings. Section 2.3 studies the automorphism of zero factor graphs of the upper triangular matrix ring of order n. Improved documentation. [In this paper, we study the automorphism of the zero factor graph of the second order matrix ring M _ 2F _ (Q) and correct the automorphism of the zero factor graph of the ring edge. [4, Theorem 3.9] and. [(5) the core error of Theorem 3.8]; In section 2.5, we study the automorphism of zero divisor graph of matrix ring MnFQ of order n. In Chapter 3, we study the automorphism group of the generalized Cayley graph GCayn (Tnn) of the upper triangular matrix semigroup. 蟽 J and 蟿. Then it is proved that any automorphism of GCayn (Tnn) can be expressed by these two automorphisms. In chapter 4, we study the automorphism group of the commutative graph 螕 M of the second order matrix ring. Firstly, we associate a contraction graph 螕 E M based on 螕 M). Then we study the relationship between Aut (螕 -M _ (+)) and Aut (螕 _ (E) M _ (+)) and find out the Aut (螕 _ (E) M _ (+)). In chapter 5, we study the automorphism of the whole graph T (螕 M _ 2N _ (2) F _ (Q)) of the second order matrix ring M _ (2) F _ (Q) when Charpy _ (F _ (Q)) 鈮，

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