关于图的自同态幺半群的研究

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,orthodox, or completely regular will be given; （2）The half-strong endomorphisms, the locally-strong endomorphisms, the quasi-strong endomorphisms and the (completely) regular endomorphisms of graphs will be characterized. The conditions under which the sets of the above endomorphisms form a monoid will be given；（3）Some algebraic properties of the strong endomorphism monoids of graphs will be explored and the monoids which can be used as the strong endomorphism monoids of some graphs will be determinded; (4) Some enumerative problems concerning the endomorphism monoids of certain graphs will be solved. In particular, the endomorphism spectra and the endomorphism type of these graphs will be given; (5) Some combinatorial characteristics which are determined by its endomorphism monoids will be given and the graph will be classified by these combinatorial characteristics. 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，目的在于建立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图的组合性质并对图进行分类。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半群、纯整半群和完全正则半群的充分必要条件；（2）刻画图的正则（完全正则）自同态、半强自同态、局部强自同态和拟强自同态，给出其构成含幺半群的充分必要条件；（3）研究图的强自同态幺半群的代数性质，确定可以作为图的强自同态幺半群的含幺半群；（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）给出图的一些可以由其自同态幺半群确定的组合特征，利用图的这些组合特征对图进行分类。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，目的在于建立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图的组合性质并对图进行分类。本项目研究了分裂图联图的正则自同态、分裂图和分裂图联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；研究了分裂图联图的拟强自同态，给出了其构成含幺半群的充分必要条件；刻画了二部图联图的自同态幺半群，给出了其是完全正则半群和逆半群的充分必要条件；刻画了二部图广义字典序积的自同态幺半群，给出了其是正则半群、纯整半群和完全正则半群的充分必要条件；研究了一类循环完全图的自同态幺半群，证明了其自同态幺半群是正则半群；研究了自同态完全正则图的性质，通过图的联、字典序积和广义字典序积给出了几类构造完全正则图的新方法。本项目的研究丰富了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。