Study On Intersection Subgroup Graphs Of Finite Groups

It is an old and innovative research field to associate a graph to a group and study the properties of a group by the structure of a setting graph.This paper mainly studies subgroup intersection graphs of a group.Let G be a finite group.The subgroup intersection graph of G is a graph whose vertices are all non-trivial subgroups of G and in which two distinct vertices H and K are adjacent if and only if H?K??e?,where e is the identity of G.In the first chapter,it mainly introduces the research history,research significance,some basic definitions and related conclusions used in this thesis,at the same time we summarize the main results of this thesis.In the second chapter,we study the structure of subgroup intersection graphs of the cyclic group,and obtain the structure theorem of subgroup intersection graphs when n having at most three prime factors,the calculation formula of automorphism group and the Wiener index of subgroup intersection graphs for an arbitrary cyclic group.Based on Chapter 2,we completely obtain integrality of subgroup intersection graphs,the classification of hyperenergetic graphs and hypoenergetic graphs for an arbitrary cyclic group in Chapter 3.In Chapter 4,we investigate the(non)oritentable genus of subgroup intersection graphs of Abelian groups.We completely characterized all Abelian groups whose subgroup intersection graphs have(non)oritentable genus is 1,2,3,4,respectively.In Chapter 5,we discuss the thickness and outerthickness of subgroup intersection graphs of Abelian groups,and the thickness and outerthickness of intersection graph of subgroups of some non-Abelian groups.We completely characterized all Abelian groups and some non-Abelian groups whose subgroup intersection graphs have thickness and outerthickness is 1,2,respectively.