图的自同态半群及惯性指数
发布时间：2018-11-02 08:42
【摘要】：代数图论是图论的重要组成部分.图的自同态半群的结构问题因深刻揭示图论和半群理论之间的关系而成为代数图论的重要研究内容C.Godsil等人在代数图论的经典教材的第六章中详细地介绍了图的自同态半群的结构与图的结构(特别是图的色数)之间的紧密联系M.Petrich和N.R.Reilly指出：正则半群(即半群中每个元素都有所谓的伪逆)由于其与群最为相像,因而在半群理论中占有重要地位.如果图的自同态半群是正则的(即图的每个自同态都有所谓的伪逆)就称该图是自同态正则的.所以研究图的自同态半群的结构,特别是研究图的自同态正则性是十分必要的.1988年,L.Marki提出了对自同态正则图做出完全分类这一公开问题.图的秩,零维数和正负惯性指数分别是指其邻接矩阵的秩,零特征值的重数和正负特征值的个数,由于图的惯性指数在化学中的应用,越来越多的学者开始研究图的惯性指数.1957年L.Collatz和U.Sinogowitz提出了刻画所有非奇异图的问题.全文共八章.第二章：研究单圈图的自同态正则性.我们证明了单圈图G是自同态正则的当且仅当G是顶点数为4,6或8的圈,或当G包含奇圈C时,每个圈外点到圈的距离至多是1,即d(G,C)≤1.此外,我们还对单圈图的联何时是自同态正则的作出完全分类；第三章：研究当树中除悬挂点外其它点的度数都相等时,树的线图的自同态正则性.本章证明了当q≥3时,q度树(即悬挂点外其它点的度数都是q)的线图L(T)是自同态正则的当且仅当树的直径不大于4；第四章：本章中我们发现了一种利用点的重化构造自同态正则图的方法.证明了当G不可回缩时,G的重化闭包Gq一定是自同态正则的,并给出了Gq的自同态谱和自同态型；第五章：本章中我们研究有限半单环及含非平凡幂等元的交换环上(理想)互极大图的自同态正则性.证明了当R是有限半单环时,其(理想)互极大图Γ'2(R)是自同态正则的,当且仅当R与以下一种环同构：Z2(?)Z2(?)Z2；F1(?)F2；M2(F),其中F, F1,F2是有限域；设R是含非平凡幂等元的有限交换环,我们证明了Γ'2(R)是自同态正则的当且仅当R与以下一种环同构：Z2(?)Z2((?)Z2；R1(?)R2,其中R1,R2是局部环;第六章：本章中我们研究群环的零因子图的自同态正则性.关于环的零因子图,D.Lu和T.Wu提出如下公开问题：对任意环R,什么情况下R的零因子图Γ(R)有正则的自同态半群?本章中对于一类重要的非交换环——群环,我们解决了上述公开问题.第七章：本章中我们研究了∞型的k圈图的零维数,证明了当k≥2时n阶∞型的k圈图的零维数集是[0,n-2k-2],并刻画了零维数是n-2k—2的极值图.第八章：本章研究了树T的线图LT的正负惯性指数.证明了ε(T)+1/2≤p(LT)≤ε(T)+1,其中ε(T)是T的内部边(非悬挂边)的个数.分别刻画了当p(LT)取上界和下界时的极值树.当p(LT)取到上界时LT是非奇异的,当p(LT)取到下界时LT是奇异的.
[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，
本文编号：2305517
[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，
本文编号：2305517
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