【摘要】：代數(shù)圖論是圖論的重要組成部分.圖的自同態(tài)半群的結(jié)構(gòu)問(wèn)題因深刻揭示圖論和半群理論之間的關(guān)系而成為代數(shù)圖論的重要研究?jī)?nèi)容C.Godsil等人在代數(shù)圖論的經(jīng)典教材的第六章中詳細(xì)地介紹了圖的自同態(tài)半群的結(jié)構(gòu)與圖的結(jié)構(gòu)(特別是圖的色數(shù))之間的緊密聯(lián)系M.Petrich和N.R.Reilly指出：正則半群(即半群中每個(gè)元素都有所謂的偽逆)由于其與群最為相像,因而在半群理論中占有重要地位.如果圖的自同態(tài)半群是正則的(即圖的每個(gè)自同態(tài)都有所謂的偽逆)就稱該圖是自同態(tài)正則的.所以研究圖的自同態(tài)半群的結(jié)構(gòu),特別是研究圖的自同態(tài)正則性是十分必要的.1988年,L.Marki提出了對(duì)自同態(tài)正則圖做出完全分類這一公開問(wèn)題.圖的秩,零維數(shù)和正負(fù)慣性指數(shù)分別是指其鄰接矩陣的秩,零特征值的重?cái)?shù)和正負(fù)特征值的個(gè)數(shù),由于圖的慣性指數(shù)在化學(xué)中的應(yīng)用,越來(lái)越多的學(xué)者開始研究圖的慣性指數(shù).1957年L.Collatz和U.Sinogowitz提出了刻畫所有非奇異圖的問(wèn)題.全文共八章.第二章：研究單圈圖的自同態(tài)正則性.我們證明了單圈圖G是自同態(tài)正則的當(dāng)且僅當(dāng)G是頂點(diǎn)數(shù)為4,6或8的圈,或當(dāng)G包含奇圈C時(shí),每個(gè)圈外點(diǎn)到圈的距離至多是1,即d(G,C)≤1.此外,我們還對(duì)單圈圖的聯(lián)何時(shí)是自同態(tài)正則的作出完全分類；第三章：研究當(dāng)樹中除懸掛點(diǎn)外其它點(diǎn)的度數(shù)都相等時(shí),樹的線圖的自同態(tài)正則性.本章證明了當(dāng)q≥3時(shí),q度樹(即懸掛點(diǎn)外其它點(diǎn)的度數(shù)都是q)的線圖L(T)是自同態(tài)正則的當(dāng)且僅當(dāng)樹的直徑不大于4；第四章：本章中我們發(fā)現(xiàn)了一種利用點(diǎn)的重化構(gòu)造自同態(tài)正則圖的方法.證明了當(dāng)G不可回縮時(shí),G的重化閉包Gq一定是自同態(tài)正則的,并給出了Gq的自同態(tài)譜和自同態(tài)型；第五章：本章中我們研究有限半單環(huán)及含非平凡冪等元的交換環(huán)上(理想)互極大圖的自同態(tài)正則性.證明了當(dāng)R是有限半單環(huán)時(shí),其(理想)互極大圖Γ'2(R)是自同態(tài)正則的,當(dāng)且僅當(dāng)R與以下一種環(huán)同構(gòu)：Z2(?)Z2(?)Z2；F1(?)F2；M2(F),其中F, F1,F2是有限域；設(shè)R是含非平凡冪等元的有限交換環(huán),我們證明了Γ'2(R)是自同態(tài)正則的當(dāng)且僅當(dāng)R與以下一種環(huán)同構(gòu)：Z2(?)Z2((?)Z2；R1(?)R2,其中R1,R2是局部環(huán);第六章：本章中我們研究群環(huán)的零因子圖的自同態(tài)正則性.關(guān)于環(huán)的零因子圖,D.Lu和T.Wu提出如下公開問(wèn)題：對(duì)任意環(huán)R,什么情況下R的零因子圖Γ(R)有正則的自同態(tài)半群?本章中對(duì)于一類重要的非交換環(huán)——群環(huán),我們解決了上述公開問(wèn)題.第七章：本章中我們研究了∞型的k圈圖的零維數(shù),證明了當(dāng)k≥2時(shí)n階∞型的k圈圖的零維數(shù)集是[0,n-2k-2],并刻畫了零維數(shù)是n-2k—2的極值圖.第八章：本章研究了樹T的線圖LT的正負(fù)慣性指數(shù).證明了ε(T)+1/2≤p(LT)≤ε(T)+1,其中ε(T)是T的內(nèi)部邊(非懸掛邊)的個(gè)數(shù).分別刻畫了當(dāng)p(LT)取上界和下界時(shí)的極值樹.當(dāng)p(LT)取到上界時(shí)LT是非奇異的,當(dāng)p(LT)取到下界時(shí)LT是奇異的.
[Abstract]:Algebraic graph theory is an important part of graph theory. The structure problem of endomorphism Semigroups of graphs has become an important part of algebraic graph theory because of revealing the relationship between graph theory and semigroup theory. C.Godsil et al introduced graph in detail in chapter 6 of classical textbook of algebraic graph theory. M.Petrich and N.R.Reilly point out that regular Semigroups (i.e. every element in a semigroup has a so-called pseudo inverse) because it is most similar to a group. So it plays an important role in semigroup theory. If the endomorphism semigroup of a graph is regular (that is, every endomorphism of a graph has a so-called pseudoinverse), the graph is called an endomorphism regular. So it is necessary to study the structure of endomorphism Semigroups of graphs, especially the regularity of endomorphism of graphs. In 1988, L.Marki put forward the open problem of complete classification of endomorphism regular graphs. The rank, zero dimension and positive and negative inertial index of graphs refer to the rank of adjacent matrix, the multiplicity of zero eigenvalues and the number of positive and negative eigenvalues respectively. More and more scholars have begun to study the inertia exponents of graphs. In 1957 L.Collatz and U.Sinogowitz proposed the problem of characterizing all nonsingular graphs. The full text consists of eight chapters. Chapter 2: the endomorphism regularity of unicyclic graphs is studied. We prove that a unicyclic graph G is an endomorphism regular if and only if G is a cycle with vertices 4 or 8, or if G contains odd cycles C, the distance from each outer point to the cycle is at most 1, that is, d (GnC) 鈮，

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