Study On Commuting Involution Graphs Of Special Class Of Finite Groups

The commuting involution graph is a graph which taking a conjugacy class of involution as its vertices,an edge exists between two vertices if and only if the two vertices are commutative.In this thesis,we first study the problem of the group structure about the 0-regular and 1-regular graphs.And then we discuss the structure of the commuting involution graph about some special groups.There are four chapters in this paper.In the first chapter,we introduce the research background of the commuting involution graph,the definition,the basic theory and the symbol.In the second chapter,we first discuss the structure and the properties of the commuting involution graph about 0-reguler and 1-reguler graph,a necessary and sufficient condition that a commuting involution graph is 0-regular is obtained.Then we prove that if the commuting involution graph)(IG(38)of the finite group G is0-regular and G(28)?I?,then G has a odd order subgroup of index 2.Finally,some examples and two guesses about 1-regular commuting involution graphs are given.In the third chapter,we discuss the structure of the commuting involution graph of the meta-cyclic 2-groups.We prove that,for a meta-cyclic 2-group,its commuting involution graphs are 0-regular or 1-regular.In the fourth chapter,we discuss the structure of the commuting involution graph of metabelian groups.For the abelian groups A and B,if G(28)AfBExt),;,(?,let?),(a be any involution of G,|{??)((10)-(28)bbH Ab?} is a subgroup of A,then the commuting involution graphs about ?),(a of G is r)1(--regular,where r is the number of all involutions which included in all of the coset ? fH(10)??),(-f??),(??)(-(10)aa,??B?.Finally,an example to calculate the structure of the commuting involution graph of a metabelian group is given.For28?(28)ZZA,22?(28)ZZB,andG(28)AfBExt),;,(?,we get all the degrees of the commuting involution of G,and show that the commuting involution graphs of G are 0-regular or 1-regular or2-regular.