The Endomorphism Monoids Of Circulant Complete Graphs K(nm,n)

The endomorphism monoids of graphs which are the connection between the composite structure of graphs and the algebraic structure, are the main research subject of algebraic graph theory. In this dissertation, the main research contents are to study the endomorphism monoids of circulant complete graphs and characterize the algebraic structure of endomorphism monoids.There are three chapters in this paper. The first chapter introduces the background and preliminary knowledge.The second chapter mainly studies the algebraic structure of endomorphism monoids of K(nm,n), get the endomorphism monoid of K(nm,n) is regular, but not orthodox semigroup. We also discuss the nature problems related to endomorphism monoids.The third chapter is an application of results in second chapter. We characterize the structure of endomorphism monoids of K(8m,8), and resolve some counting problems. In addition, we get the regularity of the endomorphism monoids of the graph K(8m,8).