Automorphism Groups Of Some Families Of Graphs And The Exchange Property Of Resolving Sets

Some structure of a graph can affect its automorphism group.Converse-ly,the automorphism group of a graph can also reveal some properties of the graph.Therefore,it is an interesting topic of studying the relationship between the properties of graphs and their automorphism groups.The resolving sets and determining sets of graphs(their exchange properties)are the basic topics which can be studied by automorphism groups of graphs.We investigated the automor-phism groups of two families of graphs(the coprime graphs and the power graphs of finite groups)which are interested in,and discussed the exchange properties of their resolving sets and determining sets,and solved the 18.49 problem in the book《unsolved problem in group theory》.In chapter II,we mainly studied the problem 18.49 in the book《Unsolved problems in group theory》:for any a,b,c,n ∈ N with 1<a,b,c ≤ n-2,then there exist α,β∈Sn such that o(α)=a,o(β)=b and o(αβ)=c,which is a conjecture proposed by Stephen Kohl in studying the Hurwitz existence problem for coverings of the projective line P1C.we firstly gave some basic properties of permutation operations,and then used these properties to give an affirmative proof of the conjecture.The coprime graph TCGn is a family of graphs related to the basic properties of integers,and so it is a interesting research topic for both graph theory scholars and number theory scholars.In chapter III,we studied some properties about the degree of vertex and vertex under the action of automorphisms,and then we used these properties to determine the automorphism group of TCGn completely.Furthermore,by using the structure of its automorphism group,we proved that TCGn is a class of IBIS graphs.At the same time,it is also pointed out that the exchange property for resolving sets in TCGn does not always holdThe power graph of a finite group is a graph which is closely related to the structure of the finite group,it is a research topic that some group theory scholars and graph theory scholars are interested in.In chapter Ⅳ,we gave the necessary and sufficient conditions of the exchange property for resolving sets in power graphs of finite groups,and then we proved that the exchange property for resolving sets in the power graph of Sn does not hold if and only if n=2+p where p is a prime number with p>3.