On Cubic Nonsymmetric Cayley Graphs

Let be a connected Cayley graph of group G, then Γ is called normal if the right regular representation of G is a normal subgroup of , the full automorphism group of Γ. For the case where G is a finite nonabelian simple group and Γ is symmetric cubic Cayley graph, Caiheng Li and Shangjin Xu proved that Γ is normal with only two exceptions. Since then, the normality of nonsymmetric cubic Cayley graph of nonabelian simple group aroused strong interest of people. So far such graphs which have been known are all normal. Then people conjecture that all of such graphs are either normal or the Cayley subset consists of involutions. In this paper we give an negative answer by two counterexamples. As far as we know these are the first examples for the non-normal cubic nonsymmetric Cayley graphs of finite nonabelian simple groups.

Normal edge-transitive Cayley graphs on a class of non-abelian groups

Let Г=Cay(G,S)be the Cayley graph of a group G with respect to its subset S.The graph is said to be normal edge-transitive if the normalizer of G in the automorphism group Aut(T)of F acts transitively on the edge set of ГIn this paper,we study the structure of normal edge-transitive Cayley graphs on a class of non-abelian groups with order 2p^(2)(p refers to an odd prime).The structure and automorphism groups of the non-abelian groups are first presented,and then the tetravalent normal edge-transitive Cayley graphs on such groups are investigated.Finally,the normal edge-transitive Cayley graphs on group G are characterized and classified.

Normality of Tetravalent Cayley Graphs of Odd Prime-cube Order and Its Application

Let p be an odd prime. In this paper we prove that all tetravalent connected Cayley graphs of order p^3 are normal. As an application, a classification of tetravalent symmetric graphs of odd prime-cube order is given.

Two sufficient conditions for non-normal Cayley graphs and their applications

A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G, S). In this paper, two sufficient conditions for non-normal Cayley graphs are given and by using the conditions, five infinite families of connected non-normal Cayley graphs are constructed. As an application, all connected non-normal Cayley graphs of valency 5 on A 5 are determined, which generalizes a result about the normality of Cayley graphs of valency 3 or 4 on A 5 determined by Xu and Xu. Further, we classify all non-CI Cayley graphs of valency 5 on A 5, while Xu et al. have proved that A 5 is a 4-CI group.

Normal edge-transitive Cayley graphs on non-abelian groups of order 4p,where p is a prime number

We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p, where p is a prime number. As a consequence we prove if IGI = 25p, δ = 0, 1, 2 and p prime, then F 1 Cay（G, S） is a connected normal 1/2 arc-transitive Cayley graph only if G = F4p, where S is an inverse closed generating subset of G which does not contain the identity element of G and F4p is a group with presentation F4p = （a, b ｜aP = b4 = 1, b-lab = a^λ）, where λ2 = -1 （mod p）.

Isomorphisms of Cubic Cayley Graphs on Dihedral Groups and Sparse Circulant Matrices

We show that,up to isomorphism,there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2n for each even number n≥4.This answers in the negative the question of Li whether all connected cubic Cayley graphs are CI-graphs(Discrete Math.,256,301-334(2002)).As an application,a formula is derived for the number of isomorphism classes of connected cubic Cayley graphs on dihedral groups,which generalises the earlier formula of Huang et al.dealing with the particular case when n is a prime(Acta Math.Sin.,Engl.Ser.,33,996-1011(2017)).As another application,a short proof is also given for a result on sparse circulant matrices obtained by Wiedemann and Zieve(arXiv preprint,(2007)).

AUTOMORPHISM GROUPS OF 4-VALENT CONNECTED CAYLEY GRAPHS OF p-GROUPS

Let G be a p-group (p odd prime) and let X = Cay(G, S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Ant(X) of X is isomorphic to the semidirect product GR x Ant(G,S), where GR is the right regular representation of G and Aut(G,S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a counterexample.

On the Classification of Arc-transitive Circulant Digraphs of Order Odd-Prime-Squared

A Cayley graph F = Cay（G, S） of a group G with respect to S is called a circulant digraph of order pk if G is a cyclic group of the same order. Investigated in this paper are the normality conditions for arc-transitive circulant （di）graphs of order p^2 and the classification of all such graphs. It is proved that any connected arc-transitive circulant digraph of order p^2 is, up to a graph isomorphism, either Kp2, G（p^2,r）, or G（p,r）[pK1], where r｜p- 1.

THE NORMALITY OF CAYLEY GRAPHS OF FINITE ABELIAN GROUPS WITH VALENCY 5

We call a Cayley digraph X = Cay(G, S) normal for G if the right regular representation of G is normal in the full automorphism group Ant(X) of X. In this paper, we give a classification of all nonnormal Cayley graphs of finite abelian groups with valency 5.

Normal Cayley graphs of finite groups

LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G, S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ?A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S) is normal. It is proved that every finite group G has a normal Cayley graph unlessG??4×?2 orG?Q 8×? 2 r (r?0) and that every finite group has a normal Cayley digraph, where Zm is the cyclic group of orderm and Q8 is the quaternion group of order 8.

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A_5

LetG be a finite group, andS a subset ofG \ |1| withS =S ?1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA 5 is a 4-Cl-group, which was a conjecture posed by Li and Praeger.

THE NORMALITY OF CAYLEY GRAPHS OF SIMPLE GROUP A_5 WITH SMALL VALENCIES

1.IntroductionLetGbeafinitegroup.ForaCayleysubsetSofGnotcontainingtheidentityelement1,theCayley(di)graphX^Cay(G,S)ofGwithrespecttoSisdefinedasthedirectedgraphwithvertexsetV(X)=GandedgesetE(X)={(g,sg)IgEG,s6S}.IfS=S--',thentheadjacencyrelationissymmet...

Finite locally primitive abelian Cayley graphs

Let Γ be a finite connected locally primitive Cayley graph of an abelian group.It is shown that one of the following holds:(1) Γ = Kn,Kn,n,Kn,n-nK2,Kn ×···× Kn;(2) Γ is the standard double cover of Kn ×···× Kn ;(3) Γ is a normal or a bi-normal Cayley graph of an elementary abelian or a meta-abelian 2-group.

Tetravalent Edge-transitive Cayley Graphs of PGL(2, p)

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.

Automorphism Groups of a Class of Cubic Cayley Graphs on Symmetric Groups

Let Sn denote the symmetric group of degree n with n 〉 3, S = {cn = （1 2…n）, cn^-1, （1 2）} and Fn=Cay（Sn, S） be the Cayley graph on Sn with respect to S. In this paper, we show that Fn （n 〉 13） is a normal Cayley graph, and that the full automorphism group of Fn is equal to Aut（Гn） = R（Sn） （Inn（C））≌- Sn×Z2, where R（Sn） is the right regular representation of Sn,φ = （1 2）（3 n）（4 n-1）（5 n-2）... （∈ Sn）, and Inn（C） is the inner isomorphism of Sn induced by φ .

On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Г=Cay(G,S)is said to be normal if G is normal in Aut Г.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Г of finite nonabelian simple groups G,where the vertex stabilizer A_(u) is soluble for A=Aut Г and v∈∨Г.We prove that either Г is normal or G=A_(5),A_(10),A_(54),A_(274),A_(549) or A_(1099).Further,11-valent symmetric nonnormal Cayley graphs of As,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.

Symmetry properties of Cayley graphs of small valencies on the alternating group A_5

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal;that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.

2p^2阶3度Cayley图(英文)

Cayley图Cay(G,S)称之为正规的,如果G的右正则表示是Cay(G,S)全自同构群的正规子群。本文决定了2p^2(p为素数)阶群上3度连通Cayley图的正规性,作为该结果的一个应用,对每一个1(?)s(?)5,对2p^2阶3度s-正则Cayley图作了分类。

3p^2阶连通4度Cayley图的正规性

对于一类3p2(p是素数)阶群G=〈a,b,c ap=bp=c3=[a,b]=1,-c 1ac=a-rbr+1,-c 1bc=a,r>1,r3≡1(m od p)〉,研究了其连通4度C ay ley图的正规性,并通过其点稳定子的结构证明G的连通4度C ay ley图均正规。鉴于王艳丽等人的相关工作,这等于圆满解决了3p2阶群的连通4度C ay ley图的正规性问题。

三类2q^2p阶群的3度Cayley图

称有限群G的Cayley图Γ是正规Cayley图,如果G的右正则表示R(G)正规于图Γ的全自同构群Aut(Γ)。研究了三类2q2p阶亚循环群的连通3度Cayley图的正规性,其中q<p均为奇素数,且q(p-1)。作为应用,决定了其中两类亚循环群的弱3-CI性。值得一提的是,在此用到单群分类定理。