Some Studies On The Endomorphism Monoids Of Graphs

This Ph.D.dissertation consists of five chapters.We study the properties and constructions of the endomorphism monoids of graphs.In chapter 3,the endomorphism monoid of(?) is explored.It is shown that End((?)) is orthodox.Some enumerative problems concerning End((?)) are solved.In particular,the endomorphism spectrum and the endomorphism type of(?) are given.In Chapter 4,End-regular(resp.,End-orthodox) graphs which are the joins of split graphs X and Y are characterized.It is proved that X+Y is never End-inverse for any split graphs X and Y.we also characterize the half-strong endomorphisms of this class of graphs.We give the conditions under which the half-strong endomorphisms of the join of split graphs form a monoid.In Chapter 5,we give several approaches to construct new End-regular(-orthodox) graphs by means of join and lexicographic product of two graphs with certain conditions. In particular,the join of two connected bipartite graphs with a regular(orthodox) endomorphism monoid is explicitly described.In Chapter 6,we characterize the endomorphism monoid of circulant complete graph K(n,3) and K(n,4).It is shown that K(n,3) is unretractive when 3(?) n and K(3m,3) is endomorphism regular.It is also shown that K(n,4) is unretractive when n = 4m + 1,4m + 3 for any m≥2,End(K(4m,4)) is regular and End(K(4m + 2,4)) is completely regular.In Chapter 7,we characterize the endomorphisms types of all generalized polygons. It is shown that for any generalized polygon X with diameter d,If d = 2,then EndotypeX=16;If d＞2,then EndotypeX = 6.