有限图的自同态幺半群的代数性质

This program is devote to the study of the algebraic theory of semigroups and graph theory. The aim of this research is try to establish the relationship between graph and its endomorphism. We will study the following five areas: （1）Some properties of the endomorphism monoids of graphs will be explored. The conditions under which the endomorphism monoids of graphs are regular will be given; （2）The regular endomorphisms and the half-strong endomorphisms of graphs will be characterized and the conditions under which the set of regular endomorphisms or the set of half-strong endomorphisms forms a monoid will be given；（3）Some End-regular graphs will be constructed by means of the join and the lexicographic product of two graphs with certain conditions；(4) Some enumerative problems concerning the endomorphism monoids of certain graphs will be solved. In particular, the endomorphism specrtra and the endomorphism type of these graphs will be given. (5) The class of graphs which are determined by its endomorphism monoids will be considered. The research of this scheme will enrich the contents of graph theory and algebraic theory of semigroups, set up some new research methods and accelerate the crossing and mutual development of graph theory and algebraic theory of semigroups.

本项目是半群代数理论和图论的交叉研究，旨在建立图的组合特征和图的自同态幺半群的代数结构之间的联系。我们将在以下五个方面展开研究：（1）研究图的自同态幺半群的代数性质，给出图的自同态幺半群是正则（纯整）半群的充分必要条件；（2）刻画图的正则自同态，给出图的正则自同态构成幺半群的充分必要条件；（3）通过图的联和字典序积，给出几类构造自同态正则图的新方法；（4）解决几类与图的自同态幺半群有关的计数问题，计算图的自同态谱和自同态型；（5） 给出一些可以由其自同态幺半群唯一确定的图类。本项目的研究将丰富半群代数理论和图论的研究内容，开辟新的研究途径，促进二者的学科交叉与共同发展。

本项目是半群代数理论和图论的交叉研究，建立了图的组合结构和图的自同态幺半群的代数结构之间的联系，利用图的自同态幺半群的代数性质研究了图的组合性质。本项目刻画了分裂图和分裂图联图的正则自同态，给出了其构成含幺半群的充分必要条件；研究了分裂图联图的拟强自同态，给出了分裂图联图的拟强自同态的具体刻画，确定了拟强自同态构成含幺半群的分裂图联图；刻画了二部图联图的自同态幺半群，给出了其是正则半群和纯整半群的充分必要条件；研究了二个图的联图的自同态完全正则性，通过图的联给出了三类构造完全正则图的新方法，确定了自同态幺半群是完全正则半群的二部图联图；研究了二个图的字典序积的自同态完全正则性，通过图的字典序积给出了一类构造完全正则图的新方法，确定了自同态幺半群是完全正则半群的二部图字典序积；刻画了分裂图的完全正则自同态，给出了其构成含幺半群的充分必要条件。本项目的研究将丰富图论和半群代数理论的研究内容，开辟新的研究途径，促进二者的交叉和共同发展。