关于图的自同态幺半群的研究

项目名称： 关于图的自同态幺半群的研究

项目编号： No.11301151

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 侯海龙

作者单位： 河南科技大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 自同态；强自同态；含幺半群；图；自同构群

英文摘要： 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

英文关键词： endomorphism；strong endomorphism；monoid；graph；automorphism group