Some Studies On The Algebraic Property Of The Endomorphism Monoids Of Graphs

The endomorphism of a graph is an edge-preserving transformation on the vertex set of a graph.The set of all endomorphism of a graph form a monoid,we call it the endomorphism monoid of graphs.The purpose of our research is to establish further relations between semigroup algebra theory and graph theory.By using the algebraic properties of endomorphism monoids of graphs,we study the combinatorial properties of graphs and classify graphs.In this paper,we mainly study the algebraic property of the endomorphism monoids of trapezoidal graph,circulant complete graph and we also studied the split graphs and the join of split graphs whose completely regular endomorphisms form a monoid.The whole paper is divided into six chapters,the thirdly to sixth chapters are the main research results of this paper.In the first chapter,we introduce the research background of this direction and theoretical significance as well as research status at domestic and overseasIn the second chapter,we give the knowledge of semigroup theory and graph theory.In the third chapter,we explore the endomorphisms;half-strong endomorphisms;local-strong endomorphisms;quasi-strong endomorphisms;strong endomorphisms;automorphism of trapezoidal graph,Using the six class of endomorphisms of trapezoidal graph,the endomorphism types of trapezoidal graph are calculated.In the fourth chapter,we study the split graph whose complete regular endomorphism,and give a specific characterization of the complete regular endomorphism of split graph,and find the sufficient and necessary condition for the split graph whose completely regular endomorphism form a monoid.The split graph of completely regular endomorphism form monoid is determined.In the fifth chapter,we study the completely regular endomorphism of the join of split graphs,and give a specific characterization of the completely regular endomorphism of the join of split graphs.We find a sufficient and necessary condition for the join of split graphs whose completely regular endomorphisms form a monoid.The join of split graph of completely regular endomorphism form monoid is determined.In the sixth chapter,we study the endomorphism monoid of a class of circulantcomplete graph.It is proved that all of them are regular semigroups,but not orthodox semigroups,and their automorphism groups are isomorphic to dihedral group of degree nm.Our research enriches the research content of semigroup algebra theory and graph theory,opens up new research approaches and research methods,and promotes the intersection and common development of these two disciplines.