本文關(guān)鍵詞：幾類代數(shù)圖的自同構(gòu)問題的研究　出處：《中國礦業(yè)大學(xué)》2016年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 零因子圖 廣義Cayley圖 交換圖 全圖 自同構(gòu)

【摘要】：代數(shù)圖論是一門應(yīng)用代數(shù)方法來解決圖論問題的學(xué)科,它促進(jìn)了代數(shù)學(xué)和圖論兩個(gè)學(xué)科的發(fā)展.在研究代數(shù)學(xué)中的非線性保持問題時(shí),由于所研究的對(duì)象為非線性映射,故研究難度很大,而研究代數(shù)圖的自同構(gòu)問題有助于研究對(duì)應(yīng)代數(shù)系統(tǒng)上的非線性保持問題,且會(huì)較容易表述并刻畫出非線性映射.所以,本論文研究代數(shù)圖的自同構(gòu)問題有著重要意義.本文主要研究了幾類代數(shù)圖的自同構(gòu)問題及n階全矩陣代數(shù)Mn(R)的由對(duì)合矩陣決定的線性映射,共分為7章.具體研究?jī)?nèi)容按照章節(jié)介紹如下第1章是緒論部分,介紹了本論文的選題意義,論文主要工作,主要研究方法和符號(hào)約定.第2章研究了矩陣環(huán)上的零因子圖的自同構(gòu)問題,其中2.3節(jié)研究了n階上三角矩陣環(huán)Tn(Fq)的零因子圖的自同構(gòu),改進(jìn)了文獻(xiàn)[3]的Tn(Fq)的環(huán)邊零因子圖的自同構(gòu)結(jié)論；2.4節(jié)研究了二階矩陣環(huán)M2(Fq)的零因子圖的自同構(gòu),糾正了[4,定理3.9]和[5,定理3.8]的核心錯(cuò)誤；2.5節(jié)研究了n階矩陣環(huán)Mn(Fq)的零因子圖的自同構(gòu),其中n≥3.第3章研究了上三角矩陣半群Tn(q)的廣義Cayley圖GCay(Tn(q))的自同構(gòu)群.構(gòu)造了GCay(Tn(q))的兩種自同構(gòu)σJ和τ,之后證明GCay(Tn(q))的任意自同構(gòu)均可由這兩個(gè)自同構(gòu)表示出來,并給出具體公式.第4章研究了二階矩陣環(huán)的交換圖Γ(M)的自同構(gòu)群.首先,基于Γ(M)我們關(guān)聯(lián)一個(gè)壓縮圖ΓE(M),然后研究Aut(Γ(M))和Aut(ΓE(M))之間的關(guān)系并求出Aut(ΓE(M)),最后刻畫Γ(M)的自同構(gòu)群.第5章研究了二階矩陣環(huán)M2(Fq)的全圖T(Γ(M2(Fq)))的自同構(gòu).當(dāng)char(Fq)≠ 2時(shí),構(gòu)造了T(Γ(M2(Fq)))的四種自同構(gòu)—τ,LP,RP,f,然后證明T(Γ(M2(Fq)))的任意自同構(gòu)均可分解成這四種自同構(gòu)的乘積,并給出具體表達(dá)式.第6章弱化了n階全矩陣代數(shù)Mn(R)的由單位積決定的線性映射(參見文獻(xiàn)[6])的限制條件,研究了Mn(R)由對(duì)合矩陣決定的線性映射.將文獻(xiàn)[6]研究的Mn(R)的保單位積線性映射ψ和在單位陣處可導(dǎo)的線性映射φ分別弱化為保對(duì)合矩陣的線性映射ψ和對(duì)合矩陣處可導(dǎo)(也即：對(duì)合矩陣處Jordan可導(dǎo))的線性映射φ,推廣了文獻(xiàn)的結(jié)論.第7章詳細(xì)總結(jié)了本學(xué)位論文的核心結(jié)論.
[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，

本文編號(hào)：1398986