Research On The Properties Of Automorphism Monoids Of Graphs

Endomorphism monoids of graphs are generalizations of automorphism groups of graphs.The aim of this research is to develop further relations between construction-s of graphs and algebraic constructions of their endomorphism monoids.We apply the property of endomorphism monoids of graphs to consider their combinatorial property and classify the graphs.This Ph.D.dissertation consists of seven chapters.Chapter 1 and Chapter 2 are preface and preliminaries respectively.In Chapter 3,we study the regular endomorphisms of trees and the join of two graphs and give conditions under which the regular endomorphisms of trees and the join of two graphs form a monoid.In Chapter 4,we consider graphs whose completely regular endomorphisms form a monoid.Trees,unicyclic graphs whose completely regular endomorphisms form a monoid are determined and the join of two graphs whose completely regular endomorphisms form a monoid are discussed.In Chapter 5,we explore the endomorphism monoids of generalized lexicographic products of bipartite graphs and give conditions under which they are regular semigroups,orthodox semigroups and completely regular semigroups.In Chapter 6,some enumerating problems about the endomorphism monoids of graphs are solved.We study the six classes of endomorphisms of P(3m + 1,3),P(3m + 2,3),P(3m,3)and fan graphs and calculate the endomorphism types or endomorphism spectrum of these graphs.Chapter 7 is conclusion and prospect for the research work.