On Locally Primitive Graphs of Order 18p?

Hua HAN Zaiping LU

1 Introduction

All graphs in this paper are assumed to be finite and simple.

LetΓbe a graph.We useV Γ,EΓand AutΓto denote the vertex set,the edge set and the automorphism group ofΓ,respectively.Then the graphΓis said to be vertex-transitive or edge-transitive if some subgroupGof AutΓ(denoted byG≤AutΓ)acts transitively onV ΓorEΓ,respectively.Recall that an arc inΓis an ordered pair of adjacent vertices.Then the graphΓis called arc-transitive if someG≤AutΓacts transitively on the set of arcs ofΓ.The graphΓis said to be locally primitive if for some subgroupG≤AutΓand eachv∈V Γ,the stabilizerGvinduces a primitive permutation groupon the neighborhoodΓ(v),the set of neighbors,ofvinΓ.For convenience,such subgroupsGare called vertex-transitive,edge-transitive,arc-transitive and locally primitive groups ofΓ,respectively.

The study of locally primitive graphs is one of the main themes in algebraic graph theory,which stems from a conjecture on bonding the stabilizers of locally primitive arc-transitive graphs(see[32,Conjecture 12]).The reader may consult[4,9–12,14,21–24,28–29,31]for some known results in this area.

In this paper,we aim at determining the arc-transitivity of certain locally primitive graphs.LetΓbe a connected graph andGbe a locally primitive group onΓ.It is easily shown thatGacts transitively onEΓ,soΓis edge-transitive.IfGis vertex-transitive,thenΓis necessarily an arc-transitive graph.Thus,for our purpose,we always assume thatΓis regular,butGis not vertex-transitive.ThenΓis a bipartite graph with two bipartition subsets being theG-orbits onV Γ.Giudici et al.[14]established a reduction for studying locally primitive bipartite graphs,which was successfully applied in[23]to the characterization of locally primitive graphs of order twice a prime power.In this paper,we concentrate our attention on analyzing the locally primitive graphs of order 18p.Our main result is stated as follows.

Theorem 1.1Let Γ be a connected regular graph of order18p,where p is a prime.Assume that Γis locally primitive.Then Γis either arc-transitive or isomorphic to one of the Gray graph and the Tutte12-cage.

2 Preliminaries

LetΓbe a graph and letG≤AutΓ.Assume thatGis edge-transitive but not vertextransitive;in this case,we callGsemisymmetric ifΓis regular.ThenΓis a bipartite graph with two bipartition subsets being theG-orbits onV Γ.Moreover,Γis arc-transitive provided thatΓhas an automorphism interchanging two of its bipartition subsets.For a given vertexu∈V Γ,the stabilizerGuacts transitively onΓ(u).Takew∈Γ(u).Then each vertex ofΓcan be written asugorwgfor someg∈G.Then two verticesugandwhare adjacent inΓif and only ifuandare adjacent,i.e.,hg?1∈GwGu.Moreover,it is well-known and easily shown thatΓis connected if and only if,=G.In particular,the next simple fact follows.

Lemma 2.1Let Γbe a connected graph and G≤ AutΓ.Assume that G is edge-transitive but not vertex-transitive.Let{u,w}be an edge of Γ.Then

(1)Guand Gwcontain no nontrivial normal subgroups in common.

(2)r≤max{|Γ(u)|,|Γ(w)|}for each prime divisor r of|Gu|.

Moreover,Γ is arc-transitive if one of the following conditions holds:

(3)G has an automorphism σof order2with=Gw.

(4)G has an abelian subgroup acting regularly on both bipartition subsets of Γ.

ProofSinceΓis connected,,=G≤AutΓ.Then(1)follows.

Letrbe a prime divisor of|Gu|withr>max{|Γ(u)|,|Γ(w)|},and letRbe a Sylowrsubgroup ofGu.ThenRfixesΓ(u)point-wise,and soR≤for each∈Γ(u).TakeQto be a Sylowr-subgroup ofGwwithQ≥R.ThenQfixesΓ(w)point-wise,and henceQ≤Gu.ThusR=Q.By the connectedness ofΓ,for eachv∈V Γ,it is easily shown thatRis a Sylowr-subgroup ofGv.ThusRfixesV Γpoint-wise,and soR=1 asR≤AutΓ.Then(2)follows.

Suppose thatGhas an automorphismσof order 2 with=Gw.Define a bijectionι:V ?！鶹 Γby(ug)ι=and(wh)ι=It is easy to check thatι∈AutΓandιinterchanges two bipartition subsets ofΓ.This implies thatΓis arc-transitive.

Suppose thatGhas a subgroupR,which is regular on both bipartition subsets ofΓ.Then each vertex inV Γcan be written uniquely asuxorwyfor somex,y∈R.SetS={s∈R|ws∈Γ(u)}.Thenuxandwyare adjacent if and only ifyx?1∈S.IfRis abelian,then it is easy to show thatux,wx,?x∈Ris an automorphism ofΓ,which leads to the arc-transitivity ofΓ.

LetGbe a finite transitive permutation group on a set Ω.The orbits ofGon the cartesian product Ω×Ω are the orbitals ofG,and the diagonal orbital{(α,α)g|g∈G}is said to be trivial.For aG-orbital Δ andα∈Ω,the set Δ(α)={β|(α,β)∈Δ}is aGα-orbit on Ω,called a suborbit ofGatα.The rank ofGon Ω is the number ofG-orbitals,which equals the number ofGα-orbits on Ω for any givenα∈Ω.AG-orbital Δ is called self-paired if(β,α)∈Δ for some(α,β)∈Δ,while the suborbit Δ(α)is said to be self-paired.For aG-orbital Δ,the paired orbital Δ?is defined as{(β,α)|(α,β)∈Δ}.Then aG-orbital Δ is self-paired if and only if Δ?= Δ.For a non-trivialG-orbital Δ,the orbital bipartite graphB(G,Ω,Δ)is the graph on two copies of Ω,say Ω×{1,2},such that{(α,1),(β,2)}is an edge if and only if(α,β)∈Δ.ThenB(G,Ω,Δ)isG-semisymmetric,whereGacts on Ω×{1,2}as follows:

If Δ is self-paired,then(α,1) ? (α,2),α∈Ω gives an automorphism ofB(G,Ω,Δ),which yields thatB(G,Ω,Δ)isG-arc-transitive.The next lemma indicates the possibility thatB(G,Ω,Δ)is arc-transitive even if Δ is not self-paired.

Lemma 2.2Let X be a permutation group onΩ,and G be a transitive subgroup of X with index|X:G|=2.LetΔbe a G-orbital.IfΔ∪Δ?is an X-orbital,then B(G,Ω,Δ)is arc-transitive.

ProofAssume that Δ∪Δ?is anX-orbital.To show thatΓ:=B(G,Ω,Δ)is arc-transitive,it suffices to find an automorphism ofΓ,which interchanges two bipartition subsets ofΓ.Takex∈XG.It is easily shown that Δx= Δ?and(Δ?)x= Δ.Define:Ω×{1,2} →Ω×{1,2};(α,1)(αx,2),(β,2)(βx,1).It is easy to check∈AutΓ,so Lemma 2.2 follows.

Moreover,the next lemma is easily shown(see also[14]).

Lemma 2.3Assume that Γ is a connected G-semisymmetric graph of valency at least2with bipartition subsets U and W,and that,for an edge{u,w}∈EΓ,two stabilizers Guand Gware conjugate in G.Then there is a bijection ι:U?W,such that Gu=Gι(u)and{u,ι(u)}EΓfor all u∈U.Moreover,Δ ={(u,ι?1(w))|{u,w}∈EΓ,u∈U,w∈W}is a G-orbital on U.In particular,ΓB(G,U,Δ),and ι extends to an automorphism of Γif and only ifΔis self-paired.

Remark 2.1LetΓandG≤AutΓbe as in Lemma 2.3.Then{Gu|u∈U}={Gw|w∈W},soGw=1 asG≤AutΓ.ThusGis faithful on both parts ofΓ.Takeu∈Uandw∈WwithGu=Gw.Thenug?wg,g∈Ggives a bijection meeting the requirement of Lemma 2.3.Thus one can definel2bijectionsι,wherelis the number of the points inUfixed by a stabilizerGu.By[7,Theorem 4.2A],l=|NG(Gu):Gu|.

LetGbe a finite transitive permutation group on Ω.LetN={x1=1,x2,···,xn}be a group of ordernlying in the centerZ(G)ofG.ThenNis normal inG,andNis semi-regular on Ω,that is,Nα=1 for allα∈Ω.Denote bytheN-orbit containingα∈Ω and bythe set of allN-orbits.ThenGinduces a transitive permutation groupon.Take a-orbitaland(,)∈.Noting that=N×Gα,it follows that={()h|h∈Gα}.Set

Then all Δi(α)are suborbits ofGatα,which are not necessarily distinct.It is easily shown thatN×Gαacts transitively on Ω1:={|1≤i≤n,h∈Gα}.It follows that allGα-orbits on Ω1have the same length divisible byFor eachi,let Δibe theG-orbital corresponding to Δi(α).

Lemma 2.4Let G,N,andΔibe as above.

(1)AllΔi(α)are suborbits of G of the same length divisible by

(2)Ifis self-paired,then for each i,there is some j,such thatΔi(α)=

(3)B(G,Ω,Δi)B(G,Ω,Δj)for1≤i,j≤n.

Proof(1)follows from the argument above the lemma.

Assume thatis self-paired.Then there is someg∈G,such that=Thus,for eachi,there are someandsuch that(α,βxi)g==.Settingwe have(α,βxi)g=Then Δi=

For eachi,de finefi:Ω×{1,2}→Ω×{1,2}byfi(δ,1)=(δ,1)andfi(δ,2)=whereδ∈Ω.It is easily shown thatfiis an isomorphism fromB(G,Ω,Δ1)toB(G,Ω,Δi).Thus(3)follows.

LetΓbe aG-semisymmetric graph.Suppose thatGhas a normal subgroupN,which acts intransitively on at least one of the bipartition subsets ofΓ.Then we de fine the quotient graphΓNto have vertices(theN-orbits)onV Γ,and twoN-orbitsBandare adjacent inΓNif and only if somev∈Band someare adjacent inΓ.It is easy to see that the quotientΓNis a regular graph if and only if allN-orbits have the same length.Moreover,ifΓNis regular,then its valency is a divisor of that ofΓ.The graphΓis called a normal cover ofΓN(with respect toGandN)ifΓNandΓhave the same valency,which yields thatNis the kernel ofGacting theN-orbits(vertices ofΓN).Thus,ifΓis a normal cover ofΓN,then the quotient groupG/Ncan be identi fied with a subgroup of AutΓN,soΓNisG/N-semisymmetric.

Corollary 2.1Let Γ and G≤ AutΓ be as in Lemma2.3.Let N≤Z(G).Then N is intransitive and semiregular on both U and W.Assume further that|N|is odd and that ΓNis the orbital bipartite graph of some self-paired orbital of,where is the subgroup ofAutΓN induced by G.Then Γ is arc-transitive.

ProofRecall thatGis faithful on bothUandW(see Remark 2.1).SinceN≤Z(G),every subgroup ofNis normal inG,soNv≤=forv∈V Γandg∈G.It follows thatNv=1,soNis semi-regular on bothUandW.Suppose thatNis transitive on one ofUandW,sayU.ThenG=NGuforu∈U,soGuis normal inGasN≤Z(G).It follows thatGufixes every vertex inU,soGu=1,which contradicts the transitivity ofGuonΓ(u).

By Lemma 2.3,there is bijectionι:U?W,such that foru∈U,the subsetι?1(Γ(u))is a suborbit ofGatu.By Remark 2.1,we may chooseι,such that it maps eachN-orbit onUto someN-orbit onW.Thusιinduces a bijectiononV ΓNinterchanging two bipartition subsetsUNandWNofΓN,whereUNandWNdenote respectively the sets ofN-orbits onUandW.Moreover,it is easily shown thatfor anyN-orbitand thatι?1(Γ(u))={|h∈Gu}for∈U,such that

Assume thatΓNis the orbital bipartite graph of some self-paired orbital of.Then,again by Lemma 2.3,∈AutΓNandis a self-paired suborbit ofat.If|N|is odd,then by Lemma 2.4,Γis isomorphic to the orbital bipartite graph of some self-paired orbital ofGonU,and henceΓis arc-transitive.

Recall that,for a groupGthat acts transitively on a set Ω,a blockBis a non-empty subset of Ω,such thatB=BgorB∩Bg=?for everyg∈G.

Lemma 2.5Let Γ be a connected graph,and let G≤AutΓ,such that G is locally primitive but not vertex-transitive.Assume that U and W are G-orbits on V Γ,and that B is a block of G on W.Then either B=W or|Γ(u)∩B|≤ 1for each u∈U.

ProofNote that for eachu∈U,eitherΓ(u)∩B=?orΓ(u)∩Bis a block ofGuonΓ(u).SinceGuacts primitively onΓ(u),we know that either|Γ(u)∩B|≤ 1 orΓ(u)?B.Suppose thatΓ(u)?Bfor someu∈U.Takew∈Bandv∈Γ(w).SinceGis edge-transitive,there isg∈Gwithvg=uandwg∈Γ(u)?B.Thenw∈Bg?1∩B,soB=Bg?1.ThusΓ(v)=(Γ(u))g?1?Bg?1=B.It follows thatΓhas a connected component with vertex setB.This yieldsB=W.

Lemma 2.6Let Γand G be as in Lemma2.5.Let U and W be the G-orbits on V Γ.Suppose that G has a normal subgroup N,which acts transitively on U.Then one of the following holds:

(1)ΓNis a|Γ(u)|-star,where u∈U.

(2)Γ is N-edge-transitive.

(3)N is regular on both U and W.

ProofIfNis intransitive onW,then(1)follows from[14,Lemma 5.5].Thus we assume thatNis transitive onW.Takeu∈U.IfNuis transitive onΓ(u),thenΓisN-edge-transitive,so(2)holds.Suppose thatNuis not transitive onΓ(u).SinceNuis normal inGuandGis locally primitive,NufixesΓ(u)point-wise.ThusNw≥Nufor eachw∈Γ(u).IfNwis transitive onΓ(w),thenΓisN-edge-transitive,so(2)holds.Thus we may further suppose thatNw≤for each∈Γ(w).By the connectedness ofΓ,we conclude thatNu=Nw=1.Then(3)follows.

Recall that a quasi-primitive group is a permutation group with all minimal normal subgroups transitive.By[14,Theorem 1.1 and Lemma 5.1],the next lemma holds.

Lemma 2.7Let Γ and G be as in Lemma2.5.Suppose that N is a normal subgroup of G,which is intransitive on both bipartition subsets of Γ.Then Γis a normal cover of ΓNand ΓNis G/N-locally primitive.If further N is maximal among the normal subgroups of G,which are intransitive on both bipartition subsets of Γ,then either ΓNis a complete bipartite graph,or G/N acts faithfully on both parts and is quasi-primitive on at least one of the bipartition subsets of ΓN.

For a finite groupG,denote by soc(G)the subgroup generated by all minimal normal subgroups ofG,which is called the socle ofG.The next result describes the basic structural information for quasi-primitive permutation groups(see[30]).

Lemma 2.8Let G be a finite quasi-primitive permutation group onΩ.Then G has at most two minimal normal subgroups,and one of the following statements holds:

(1)andsoc(G)is the unique minimal normal subgroup of G,whered≥1and p is a prime;in this case,G is primitive onΩ.

(2)soc(G)=Tlfor l≥1and a nonabelian simple group T,and eithersoc(G)is the unique minimal normal subgroup of G,orsoc(G)=M×N for two minimal normal subgroups M and N of G with|M|=|N|=|Ω|.

3 The Quasi-primitive Case

LetΓbe aG-locally primitive regular graph of order 18p,whereG≤AutΓandpis a prime.Assume thatGis intransitive onV Γ.ThenΓis a bipartite graph with two bipartition subsets beingG-orbits,sayUandW.

Assume next thatGacts faithfully on bothUandW,and thatGis quasi-primitive on one ofUandW.IfGacts primitively on one ofUandW,then by[18],Γis either arc-transitive or isomorphic to one of the Gray graph and the Tutte 12-cage.Thus we assume in the following that neitherGUnorGWis a primitive permutation group.Then by Lemma 2.8,N:=soc(G)is the direct product of isomorphic non-abelian simple groups.In particular,Gis insoluble,soΓis not a cycle.

Without loss of generality,we assume thatGis quasi-primitive onU.Recall thatGUis not primitive.Take a maximal blockBU)ofGonU.Then|B|is a proper divisor of|U|=9pandfor eachuThenandGacts primitively onB.SinceGis quasi-primitive onU,we know thatGacts faithfully onB.Thus we may viewGas a primitive permutation group(onB)of degree

Lemma 3.1|B|=3or9.

ProofIt is easy to see that|B|=3,9 orp.Suppose that|B|=p.Then|B|=9 and by[7,Appendix B],N=soc(G)=A9or PSL(2,8).IfN=A9thenNBA8andp≤7;however,A8has no subgroups of indexp,a contradiction.ThusN=PSL(2,8),andp=7,soand|U|=63,whereu∈B.SinceΓisG-locally primitive,Guinduces a primitive permutation groupIfG=N,thenyielding thatΓis a cycle,a contradiction.It follows thatG=PΣL(2,8)PSL(2,8)and|Gv|=24,wherevis an arbitrary vertex ofΓ.Checking the subgroups of PSL(2,8)in the Atlas(see[6]),we know thatNhas no proper subgroups of index dividing 21.It implies thatNis transitive onW,soGis also quasi-primitive onW.By the information given for(2,8)in[6],for eachv∈V Γ.Then eitherandΓis cubic,orandΓhas valency 4.Take{u,w}∈EΓ.ThenIt follows thatGuandGwhave the same center,which contradicts Lemma 2.1.

Therefore,Gis a primitive permutation group(onB)of degreepor 3p.For further argument,we list in Tables 1–2 the insoluble primitive groups of degreepand of degree 3p,respectively.Noting thatNBhas a subgroup of index|B|=9 or 3,it is easy to check thatN=A6or PSL(n,q).Suppose thatN=A6.Then|B|=3 andp=5.It follows thatGuis a 2-group.SinceΓisG-locally primitive,

ThenΓis a cycle,a contradiction.Thus the next lemma follows.

Table 1 Insoluble transitive groups of prime degree(see[2,Table 7.4])

Table 2 Insoluble primitive groups of degree 3p(see[16])

Lemma 3.2Either|B|=9and N=PSL(n,q)with n prime,or|B|=3and N=PSL(3,q)with q≡1(mod 3).

Letbe the Galois field of orderq,and letbe then-dimensional linear space of row vectors overDenote byPandH,respectively,the sets of 1-subspaces and(n?1)-subspaces ofThen the action ofN=SL(n,q)/Z(SL(n,q))onBis equivalent to one of the actions ofNonPand onHinduced by

whereA=(aij)n×n∈SL(n,q).Letσbe the inverse-transpose automorphism of SL(n,q),that is,

Thenσgives an automorphism ofNof order 2.Define

Then

For 1≤i≤n,leteibe the vector with theith entry 1 and other entries 0.Then

where

For a subgroupXof SL(n,q),we denoteto be the image ofXinN,that is,=X/Z(SL(n,q)).Then the following lemma holds.

Lemma 3.3If B∈B,then NBis conjugate in N to one

The following simple fact may be shown by simple calculations.

Lemma 3.4Set

Thenacts transitively onand has two orbits on H with lengthsrespectively.Moreover,for each divisor m of q?1,Q:H has a unique subgroup containing Q:L and having index m,which is

Lemma 3.5Write q=rffor a prime r and an integer f≥1.Assume that|B|=9for B∈B.Then the following statements hold:

(1)(n,q)(2,2),(2,3),(3,2),(3,3).

(2)n is an odd prime with q1(modn).

(3)n is the smallest prime divisor of nf.

ProofBy Lemma 3.2,N=PSL(n,q)for a primen.Since 9 is a divisor of|N|andNis simple,(n,q)(2,2),(2,3),(3,2).

Suppose thatN=PSL(3,3).Thenp=13,G=N,|GB|=24·33and|Gu|=48.Takew∈Γ(u).SinceΓis regular,|Gu|=48=|Gw|.Checking the subgroups of SL(3,3)(see[6]),we haveSinceΓisG-locally primitive,andΓhas valency 4.ThusGuwD12.It follows thatGuandGwhave the same center isomorphic towhich contradicts Lemma 2.1.Thus(1)follows.

Suppose thatn=2.Then,sinceis a prime,r=2 andf=2sfor some integers≥1.Thusand henceBut 22s?1 is not divisible by 9,a contradiction.This implies thatnis an odd prime.Ifq≡1(modn),thena contradiction.Then(2)follows.

Ifnf=6 andr=2,thenp==21 or 63,a contradiction.Thus,by Zsigmondy’s theorem(see[20,p.508]),there is a prime,which dividesrnf?1,but notri?1 for all 1≤i≤nf?1.Clearly,such a prime isp.Suppose thatfhas a prime divisors,such thats

Lemma 3.6Let B∈B.If(n,q)=(3,8),then|B|=9and Γ is arc-transitive and of valency8or64.

ProofAssume that(n,q)=(3,8).ThenNSL(3,8),p=73 and|G:N|=1 or 3.By Lemma 3.2,|B|=9.Without loss of generality,we assume thatN=SL(3,8)and chooseB,such thatNB=P:H,whereandHis defined above Lemma 3.3.

SinceNBis transitive onB,it is easily shown thatPacts trivially onB,soHacts transitively onB.Then|H:Hu|=9.Note thatChecking the subgroups of PSL(2,8),we conclude that the action ofHonBis equivalent to the action ofHon the set of 1-subspaces ofThen,without loss of generality,we may assume thatHuis conjugate to

Recall that a(1,2)-flag ofis a pair{V1,V2}of a 1-subspace and a 2-subspace with the 1-subspace contained in the 2-subspace.SinceP≤Nu,we haveIt is easily shown thatNuis the stabilizer of some(1,2)-flag{V1,V2}inN.It follows that the action ofNonUis equivalent to the action ofNon the setFof(1,2)-flags of

Now we show that the actions ofNonUandWare equivalent.Note that|G:N|=1 or 3.Thus,sinceWis aG-orbit,eitherNis transitive onWorNhas 3 orbits onW.Checking the subgroups of SL(3,8),we know thatNhas no subgroups of index 219.It follows thatNis transitive onW.Note thatN=SL(3,8)has no subgroups of index 3,9 or 219.It follows that a maximal block ofNonWhas size 9.Then a similar argument as above implies that the action ofNonWis also equivalent to that onF.Moreover,ΓisN-edge-transitive by Lemma 2.6.

IdentifyingUwithF,and by Lemma 2.3,ΓB(N,F,Δ),where Δ is anN-orbital onF.Without loss of generality,chooseuto be the flagCalculation shows that Δ(u)is one of the following 5 suborbits:

(i)which are self-paired and of length 23.

(ii)andwhich are paired to each other and of length 26.

(iii)which is self-paired and of length 29.

Suppose that Δ(u)is the suborbit in(iii).ThenΓhas valency 29.Recall that|G:N|=1 or 3,andIt follows thatGu/Nuis cyclic,and henceGuis soluble.SinceΓisG-locally primitive,is a soluble primitive permutation group of degree 29.In particular,socand socis the unique minimal normal subgroup ofThusasNuinduces a normal transitive subgroup ofHowever,the unique Sylow 2-subgroup ofNuis non-abelian and has order 29,a contradiction.

If Δ(u)is described in(i),thenΓhas valency 8,and by Lemma 2.3,Γis arc-transitive.

Assume that Δ(u)is one of the suborbits in(ii).ThenΓhas valency 64.Letσbe the inverse-transpose automorphism ofN=SL(3,8).ThenFisσ-invariant.Consider the actionNonFand takea∈SL(3,8)with

Thenwhich interchanges the two suborbits in(ii). It follows from Lemma 2.2 thatΓis arc-transitive.

Lemma 3.7Assume that(n,q)(3,8).Then there is u∈U withwhere σ is the inverse-transpose automorphism ofSL(n,q),and and are described as in Lemmas3.3and3.4,respectively.In particular,q≡1(mod|B|).

ProofRecall that the action ofN=SL(n,q)/Z(SL(n,q))onBis equivalent to one of the actions ofNonPand onH.Without loss of generality,we may chooseB∈B,such thatwhereandis described as in Lemma 3.3.Setq=rffor some primerand integerf≥1.ThenRis a nontrivialr-group.

Takeu∈B.Then|NB:Nu|=|B|=3 or 9.Suppose thatNoting thatRNuis a subgroup ofNBasRis normal inNB,it follows thator 9.In particular,Ris a 3-group,and hence|B|=9 by Lemma 3.2.Then,by Lemma 3.5,nandq?1 are coprime,soZ(SL(n,q))=1.ThusNSL(n,q)andAssume that|(RNu):Nu|=9.ThenNB=RNu.It implies thatR∩Nuis normal inNB.ThenyieldingR∩Nu=1.It follows thatBy Lemma 3.5,we conclude thatn=3 andf=1,that is,(n,q)=(3,3),a contradiction.Thus|NB:(RNu)|=3.Noting thatit follows that GL(n?1,3f)has a subgroup of index 3.Note thatwheredis the largest common divisor ofn?1 and 3f?1.It implies that PSL(n?1,3f)has a subgroup of index 3.Thenn=3 andf=1,a contradiction.Therefore,Ris contained inNu.

Sinceis normal inNB,we know thatis a subgroup ofNB.Suppose thatThenLetZbe the center of Thenanddivides 9.By Lemmas 3.2 and 3.5,n≥3 and(n,q)(3,2),(3,3).Thusis simple,and hence it has no subgroups of order 3.Suppose thatThenhas a primitive permutation representation of degree 9.By[7,Appendix B],we conclude thatThen(n,q)=(3,8),a contradiction.It follows thatthat is,Consider the commutator subgroups ofLand.By[19,Chapter II,Theorem 6.10],=L,and hencea contradiction.Therefore,the first part of this lemma follows.

LetXandYbe the pre-images ofNBandNuin SL(n,q).ThenMoreover,X=Q:HorQσ:HandY≥Q:LorQσ:L,respectively.It follows that|B|is a divisor of|H:L|=q?1.Thenq≡1(mod|B|).

Theorem 3.1Γ is an arc-transitive graph,and one of the following holds:

(1)N=PSL(3,8),p=73and Γ has valency8or64.

(2)N=PSL(n,q),and Γhas valency qn?1,where q≡1(mod 9),n≥5and(n,q)satisfies Lemma3.5.

(3)N=PSL(3,q),3p=q2+q+1and Γhas valency q2,where q≡1(mod 3).

ProofBy Lemmas 3.2 and 3.5,N=soc(G)=PSL(n,q)for some odd primen.If(n,q)=(3,8),then(1)follows from Lemma 3.6.Thus we assume that(n,q)(3,8)in the following.Writeq=rffor a primerand an integerf≥1.

Case 1Assume that|B|=9.Thenis a prime.By Lemma 3.7,q≡1(mod 3),soIt follows thatn3.By Lemmas 3.5,nfhas no prime divisors less than 5.Note that|G:N|dividesnfandGis transitive onW.It follows that the number ofN-orbits onWis a divisor ofnf.It implies thatNis transitive onW,and henceGis quasi-primitive onW.

Recall thatGis faithful and imprimitive onW.Take a maximal blockCofGonW,and setThenGacts primitively onC.

Sincen≥5,checking Table 3,we conclude thatGhas no primitive permutation representation of degree 3p.Then|C|3.In addition,Ghas no subgroups of index 9,so|C|p.It follows that|C|=9 and|C|=p.Then the argument for the actions ofNonBandUis available for the actions onCandW.This allows us to viewBas a copy ofP,and to viewCas a copy ofPorH.

ChooseB∈BandC∈C,such thatandThen,by Lemmas 3.4 and 3.7,andwhereu∈B,w∈CandXis a subgroup of SL(n,q)consisting matrices of the following form:

Note thatΓisG-locally primitive andNis not regular on bothUandW.By Lemma 2.6,ΓisN-edge-transitive.ThenΓ(u)is anNu-orbit onW.Thus,for anNu-orbitonC,either

Suppose thatNC=NB.Then bothBandCcorrespond toBy Lemma 3.4,for eachu∈B,the stabilizerNuis transitive onC{C}.Thus eitherNote thatNufixesCpoint-wise asNu=Nwis normal inNB=NC,wherew∈C.ThenChoose∈Ccorresponding toand take∈LetY1andY2be the pre-images ofNu∩andNu∩,respectively.Then

It follows that|(Nu∩):(Nu∩)|=|Y1:Y2|=|9|=soNu∩is transitive onThen?Γ(u),which contradicts Lemma 2.5.

Now letThenBandCcorrespond toand|2≤i≤respectively.By Lemma 3.4,Nuhas two orbitsC1andC2onC,whereC1has lengthand containsC1corresponding to|1≤i≤n?andC2has lengthqn?1and containsC2corresponding toCalculation shows thatwherew1∈C1andw2∈C2.IfΓ(u)then we get a similar contradiction as above.ThusandΓ(u)is one of the 9-orbits ofNuonNote thatforw∈C2.ThenΓis arc-transitive by Lemma 2.1,and(2)follows.

Case 2Assume that|B|=3.ThenN=PSL(3,q)withq=rf≡1(mod 3).In particular,|N|has at least 4 distinct prime divisors(see[15,p.12]).

LetW1be an arbitraryN-orbit onW.Takew∈W1.Thenor 9p.SinceNis simple,Nhas no subgroups of index 3.By[7,Appendix B],Nhas no subgroups of index 9.By Table 3,Nhas no primitive permutation representations of prime degree,soNhas no subgroups of indexp.Thus|W1|=3por 9p.Suppose that|W1|=3p.ThenNhas exactly three orbits onW.SinceNis normal inG,eachN-orbit onWis a block ofG.By Lemma 2.5,for some integerss,t≥0.ThenThus|N|has at most 3 distinct prime divisors,a contradiction.Then|W1|=9p,that is,Nis transitive onW.

Take a maximal blockCofGonW,and setThenGacts primitively onC.Recall thatNhas no subgroups of index 3,9 orp.It implies that|C|=3p.Then(3)follows from an analogous argument given in Case 1.

4 The Proof of Theorem 1.1

LetΓbe aG-locally primitive regular graph of order 18p,whereG≤AutΓandpis a prime.Assume thatGis intransitive onV Γ.LetUandWbe theG-orbits onV Γ.IfGacts unfaithfully on one ofUandW,thenΓis the complete bipartite graph K9p,9p,and henceΓis arc-transitive.Thus we assume thatGis faithful on bothUandW.By the argument in Section 3,we assume further thatGhas non-trivial normal subgroups,which are intransitive on bothUandW.LetMbe maximal in such normal subgroups ofG.Denote byandthe sets ofM-orbits onUandW,respectively.For eachv∈V Γ,denote bytheM-orbit containingv.

By Lemma 2.7,Γis a normal cover ofΓM.ThenMis semi-regular on bothUandW;in particular,|M|=3,9,por 3pandp,9 or 3,respectively.Note thatMis the kernel ofGacting onThen we identifyX:=G/Mwith a subgroup of AutΓM.ThenΓMisX-locally primitive.

Next we finish the proof of Theorem 1.1 in two subsections depending on whether or notΓMis a bipartite complete graph.

4.1 Graphs with ΓMcomplete bipartite

In this subsection,we assume thatΓMis a complete bipartite graph,that is,Letu∈Uandw∈W.Thenandact primitively onand,respectively.ThusXacts primitively on bothand.Moreover,sois a divisor of|X|. foru∈U,yielding|Γ(u)|≤3.By Lemma 2.1,

Lemma 4.1Assume that X is faithful on one ofand.Then Γ is an arc-transitive graph of order36and valency6.

ProofWithout loss of generality,we may assume thatXis faithful on.Then bothXandare primitive permutation groups on.If|M|=3p,thenor S3,and henceXis intransitive on the edges ofΓM,a contradiction.If|M|=9,thenp2is a divisor ofX;however,each permutation group of degree primephas order indivisible byp2,a contradiction.If|M|=p,then soc(X)and socare one of A9,PSL(2,8)oryieldinga contradiction.

Now let|M|=3.ThenandSince 9p2is a divisor of|X|,checking Table 3 implies that soc(X)=A3por A5.Note thatandIt follows thatsoc(X)A6and soc()A5.

By soc(X)～=A6,we know thatXis isomorphic to a subgroup of AutIn particular,|X:soc(X)|is a divisor of 4.Since soc(X)is normal inX,all soc(X)-orbits onhave the same length dividing 3p.Thus the number of soc(X)-orbits onis a common divisor of 4 and 3p.It follows that soc(X)acts transitively on.In addition,soc(X)is transitive,asXis faithful and primitive on.ThenΓMis soc(X)-edge-transitive by Lemma 2.6.In particular,socand socact transitively onand,respectively.Checking the subgroups of A6,we conclude that socand socand socare not conjugate in soc(X).It is easy to see thatΓis soc(X)-locally primitive.

LetHbe the pre-image of soc(X)inG.ThenH=M.soc(X),M=Z(H)andΓisH-locally primitive.Letbe the commutator subgroup ofH.Suppose thatH.ThenH=M×andA6.Thusis normal inHand intransitive on bothUandW.By Lemma 2.7,is semi-regular onV Γ,which is impossible.Therefore,H=.By the information given in[6],we know thatHhas an automorphismσof order 2 withfor suitableandNoting thatandfor arbitraryit follows thatThen,by Lemma 2.1,Γis an arc-transitive graph.

Lemma 4.2Assume that X acts unfaithfully on bothand.Then Γ has valency2,3or p,and Γ is either arc-transitive or isomorphic to the Gray graph.

ProofLetY1andY2be the corresponding kernels.ThenY1∩Y2=1 andY1Y2=Y1×Y2.SinceXacts primitively on bothand,we conclude thatY1andY2act transitively onand,respectively.It follows that socwherei=1,2.Checking primitive permutation groups of degreewe conclude thatY1×Y2contains a normal subgroupY=T1×T2,which is transitive onEΓM,such thatand one of the following conditions holds:

(v)|M|=3 orp,T1=soc(Y1)T2=soc(Y2)andT1is non-abelian simple.

LetNbe the pre-image ofYinG.ThenΓisN-edge-transitive.In particular,Nis not regular onUandW.Noting thatNis faithful on bothUandW,it follows thatNis not abelian.

If(i)occurs,thenΓMis a cycle,soΓis arc-transitive.

Assume that(ii)occurs.ThenYhas a subgroup,which has orderpand acts regularly on bothand.ThusNhas a subgroupacting regularly on bothUandW.By Sylow’s theorem,it is easily shown thatIt follows from Lemma 2.1 thatΓis vertex-transitive,and henceΓis arc-transitive.

Assume that(iii)occurs.Then|M|=3pandIfp=3,then eitherΓis arc-transitive or by[26]or[27],Γis isomorphic to the Gray graph.Assume thatp=2.ThenMhas a characteristic subgroupand henceKis normal inN.It is easily shown thatΓis a normal cover ofΓKwith respect toNandK.ThusΓKis a cubic edge-transitive graph of order 12.However,by[3,5],there are no such graphs,a contradiction.Thus assume thatp≥5.ThenMhas a unique Sylowp-subgroup.LetPbe the unique Sylowp-subgroup ofM.ThenandPis normal inN.SinceΓis cubic,ΓisN-locally primitive.ThusΓis a normal cover ofΓP,and henceΓPis anN/P-edge-transitive cubic graph of order 18.WriteN=P:Q,whereQis a Sylow 3-subgroup ofN.Thenis non-abelian.

LetSbe the Sylow 3-subgroup of CN(P).ThenSis normal inN.It is easily shown thatSfixes bothUandWset-wise,soSis intransitive on bothUandWas|U|=|W|=9pandp3.ThenSis semi-regular on bothUandW,so|S|=1,3 or 9;in particular,Sis ablelian.It implies thatPS=P×Sis abelian and semi-regular on bothUandW.Assume|S|=3.SinceSis normal inQ,it implies thatSlies in the center ofQ.Note thatThenQ/Sis cyclic.It follows thatQis abelian,a contradiction.Therefore,|S|=9,and hencePSis regular on bothUandW.ThusΓis arc-transitive by Lemma 2.1.

Next we finish the proof by excluding(iv)–(v).

Suppose that(iv)occurs.WriteN=P:Q,whereQis a Sylow 3-subgroup ofN.ThenLetSbe the Sylow 3-subgroup of CN(P).ThenSis normal inN.SinceNis non-abelian,QS.Consider the quotientN/CN(P).We conclude thatSinceΓis bipartite,it is easily shown thatSfixes the bipartition ofΓ.Ifp3,thenSis neither transitive nor semi-regular on bothUandW,which contradicts Lemma 2.7.Thusp=3,so|V Γ|=54 and|AutΓ|is divisible by 35.By[3,5],there exists no such cubic edge-transitive graph,a contradiction.

Suppose that(v)occurs.Note thatorSinceY=N/Mis the direct product of two isomorphic non-abelian simple groups,it follows thatN/M=CN(M)/M,soN=CN(M).ThenMis the center ofN.Takeu∈U.ThenThenNuacts transitively on,and henceacts transitively onW.Note thatNuhas a normal subgroupwhich acts trivially on.ThenKfixes set-wise eachM-orbit onW.It is easily shown thatKis normal inIt follows that allK-orbits onWhave the same length.Thus eitherKacts trivially onW,orKacts transitively on eachM-orbit onW.The latter case implies thata contradiction.ThusK=1 asGis faithful on bothUandW,soNoting thatT2is transitive on,it follows thator 3p,which contradicts thatT2is simple.

4.2 Graphs with ΓMnot complete bipartite

Now we assume thatΓMis not a complete bipartite.ThenXacts faithfully on bothand.By Lemma 2.7,Xis quasi-primitive on one ofandRecall that3p,p,9 or 3.

Lemma 4.3

ProofSuppose thatWithout loss of generality,we assume thatXis quasiprimitive on.Then it is easily shown thatXis primitive on.Thus soc(X)is isomorphic to one of A9,PSL(2,8)orLetN≤GwithN/M=soc(X).

Assume that soc(X)PSL(2,8).ThenXis 3-transitive on bothand.It follows thatand thatΓisN-locally primitive.Moreover,it is easily shown thatMis the center ofN.By[6],PSL(2,8)has Schur multiplier 1.This implies thatN=M×KwithThusNhas a normal subgroupKacting neither transitively nor semi-regularly on each ofUandW,which contradicts Lemma 2.7.

Assume that socA similar argument as above implies thatandΓisN-locally primitive.Moreover,Nis a central extension ofMby A9.Ifp2,then noting that A9has Schur multiplierwe haveN=M×KforK

Assume that socThenXAGL(2,3),and for somethe stabilizeris isomorphic to an irreducible subgroup of GL(2,3).By[13,Theorem 2],there are no semisymmetric graphs of order 18.It follows from[17,Lemma 2.5]that soc(X)acts transitively onThus soc(X)is regular on bothandBy[25],acts faithfully on the neighbors ofIn addition,sinceΓMisX-locally primitive,is a primitive permutation group onHowever,it is easy to check that GL(2,3)has no irreducible subgroups satisfying the conditions fora contradiction.

Lemma 4.4or p,then Γ is arc-transitive.

ProofIf=2,thenandΓMis 4-cycle,which is impossible.IfthenandΓMis 6-cycle,and henceΓis a cycle.Thus we assume thatThen|M|=9,and eitherorXis a permutation group with soc(X)listed in Table 3.In particular,Ghas a subgroupwhich acts regularly on bothUandW.By Sylow’s theorem,it is easily shown thatwherePis a Sylowp-subgroup ofR.ThenRis abelian,and henceΓis arc-transitive by Lemma 2.1.

Finally,we deal with the case wherethat is,p3 and

Lemma 4.5Assume that=3p9.Then Γ is arc-transitive.

ProofWithout loss of generality,we assume thatX=G/Mis a quasi-primitive group onSinceby Lemma 2.8,soc(X)is insoluble.

Case 1Assume thatX=G/Mis primitive onThenXis known as in Table 2.Since soc(X)is non-abelian simple,it has no proper subgroups of index less than 5.Suppose that soc(X)is not primitive onThen either each soc(X)-orbit onhas lengthp,or soc(X)is transitive onwith a block of size 3;moreover,p>3 in both cases.Thus,for these two cases,soc(X)can be viewed as a transitive permutation group of prime degree.Checking Tables 1–2,we conclude that socand socwhereαis either anM-orbit onor a block of soc(X)with size 3 onFor the former case,soc(X)α|≤|S7:A6|=14,a contradiction;for the latter case,A6has a subgroup of index 3,which is impossible.It follows that soc(X)is primitive on bothand;in particular,ΓMis soc(X)-edge-transitive.

LetN≤GwithN/M=soc(X).Clearly,Nis normal inGandΓisN-edge-transitive.Moreover,it is easily shown thatMis the center ofN.

Subcase 1.1Assume that the extensionN=M.soc(X)splits overM,that is,N=M×Kfor socThenKis a normal subgroup ofG,andKacts primitively on bothand.SinceKis a non-abelian simple group,its order has at least three distinct prime divisors.It follows thatKis not semi-regular on bothUandW.ThenKis transitive on one ofUandW.This implies that 9pis a divisor of|K|,soKis not isomorphic to one of A5,PSL(3,2)and PSL(2,2f).

Without loss of generality,assume thatKis transitive onU.Then,foru∈U,the stabilizeris transitive on theM-orbitThussoKhas a subgroup of index 3.Noting thatit implies thatChecking the subgroups of socwe know that eitherandp=5,orPSL(3,q)and 3p=q2+q+1,whereqis a power of a prime withq≡1(mod 3).

Assume soc(X)=A6.ThenΓhas order 90.Suppose thatKis intransitive onW.ThenKhas three orbits onW,soΓis cubic by Lemma 2.6.ThusΓis a semisymmetric cubic graph by[5,Theorem 5.2].Again by[5],there is no semisymmetric cubic graphs of order 90,a contradiction.ThenKis also transitive onW.By Lemma 2.6,ΓisK-edge-transitive.Checking the subgroups of A6,we know thatKuD8foru∈U.It follows thatΓhas valency 4 or 8.SinceΓisG-locally primitive,is a primitive group of degree 4 or 8.Sinceis a transitive normal subgroup of,it follows thatΓhas valency 4.ThenΓMhas valency 4.Consider the actions of soc(X)onandIf these two actions are equivalent,thenΓMhas valency 6 or 8;otherwise,ΓMhas valency 3 or 12.This is a contradiction.

Assume that soc(X)=PSL(3,q).ThenΓMhas valencyq2,q+1 orq2+q.IfKis intransitive onW,thenKhas three orbits onW,and henceΓis cubic by Lemma 2.6,a contradiction.ThusKis also transitive onW,soΓisK-edge-transitive.Arguing similarly as in the proof of Theorem 3.1,we conclude thatΓis arc-transitive and has valencyq2.

Subcase 1.2Assume that the extensionN=M.soc(X)does not split overM.Then checking the Schur multipliers of the simple groups in Table 3,conclude thatN=3.A6withp=5 or 2,orN=3.A7withp=5 or 7,orN=SL(3,q)with 3q?1.

LetN=SL(3,q)with 3q?1.Using the notation defined above in Lemma 3.3,we identifywithPandwithPorH.Then there areandsuch that

andBy Lemma 3.4 and a similar argument as in the proof of Theorem 3.1,it is easily shown thatΓis an arc-transitive graph of valencyq2.

LetN=3.A6.Ifp=2,thenand henceΓis arc-transitive by Lemma 2.4.Now letp=5.ThenΓMhas valency 6,8,3 or 12.Takeu∈U.ThensoSinceΓisG-locally primitive,is a primitive group.Noting thatis a transitive normal subgroup ofit follows thatΓhas valency 4 or 3.SinceΓis a normal cover ofΓM,we conclude thatΓhas valency 3.By[5],there is no semi-symmetric cubic graphs of order 90.ThusΓis arc-transitive.

LetN=3.A7withp=5 or 7.Assume first that soc(X)acts equivalently onandThen by Lemma 2.3,ΓMis isomorphic to an orbital bipartite graph of soc(X)onCalculation shows that the suborbits of soc(X)onare all self-paired.ThenΓis arc-transitive by Colloray 2.1.Ifp=5,thenX=soc(X)PSL(2,7)andΓMhas valency 14;however,PSL(2,7)has no primitive permutation representations of degree 14,a contradiction.Thenp=7.It is easily shown thatΓhas valency 10.

Assume that the actions of soc(X)onandare not equivalent.ThenX=soc(X)=A7andPSL(2,7),soG=N=3.A7.In particular,p=5 andΓMhas order 30.TakeChecking the subgroups of A7,we conclude that=7 or 8.ThenΓMhas valency 7 or 8,and so doesΓ.Veri fied by GAP,there are two involutionssuch thatforv∈V Γ.Thus we may choose a suitablesuch thatfor an automorphism ofGof order 2.ThenΓis arc-transitive by Lemma 2.1.

Case 2 Assume thatX=G/Mis quasi-primitive,but not primitive onLetBbe a maximal block ofXonThen|B|=3.SetB=Then|B|=pandXacts faithfully onB.ThusXis known as in Table 3.LetThen|XB:Xu|=|B|=3.Checking one by one the groups listed in Table 3,we conclude that soc(X)=PSL(n,q)with

Suppose thatn=2.Thenq=for some integers≥1,andN=M.soc(X)×PSLIt follows thatGhas a normal subgroupKisomorphic to PSLNote that 9 is not a divisor of|K|.It follows thatKis intransitive on bothUandW.By Lemma 2.7,Kis semi-regular onU,which is impossible.Thenn≥3.

A similar argument as above implies that(n,q)(3,2).Then by[15,p.12],|soc(X)|has at least four distinct prime divisors.Noting|X|=it follows thathas an odd prime divisor other than 3.This implies that the valency ofis no less than 5.If soc(X)is intransitive onthen soc(X)has exactly three orbits onsoΓMhas valency 3 by Lemma 2.6,a contradiction.Therefore,soc(X)is transitive onand henceΓMis soc(X)-edge-transitive.LetN≤GwithN/M=soc(X).ThenNis normal inGandΓisN-edge-transitive.

It is easily shown thatnis an odd prime with(modn)(see the proof of Lemma 3.5).Then the Schur multiplier of PSL(n,q)is 1.Recallingit yields thatN=M×K,whereClearly,Kis a normal subgroup ofG.Recalling that soc(X)is transitive on bothandwe conclude that eachK-orbit onV Γhas length at least 3p.SinceKis not semi-regular andΓhas valency no less than 5,by Lemma 2.6,we know thatΓisK-edgetransitive.Then the argument in Section 3 implies thatΓis an arc-transitive graph.

AcknowledgementThe authors would like to thank the referees for valuable comments and careful reading.Note that=N×Gvand

[1]Aschbacher,M.,Finite Group Theory,Cambridge University Press,Cambridge,1993.

[2]Cameron,P.J.,Permutation Groups,Cambridge University Press,Cambridge,1999.

[3]Conder,M.and Dobcsányi,P.,Trivalent symmetric graphs on up to 768 vertices,J.Combin.Math.Combin.Comput.,40,2002,41–63.

[4]Conder,M.D.,Li,C.H.and Praeger,C.E.,On the Weiss conjecture for finite locally primitive graphs,Proc.Edinburgh Math.Soc.(2),43,2000,129–138.

[5]Conder,M.,Malni?,A.,Maru?i?,D.and Poto?nik,P.,A census of semisymmetric cubic graphs on up to 768 vertices,J.Algebr.Comb.,23,2006,255–294.

[6]Conway,J.H.,Curtis,R.T.,Noton,S.P.,et al.,Atlas of Finite Groups,Clarendon Press,Oxford,1985.

[7]Dixon,J.D.and Mortimer,B.,Permutation Groups,Springer-Verlag,New York,Berlin,Heidelberg,1996.

[8]Du,S.F.and Xu,M.Y.,A classification of semisymmetric graphs of order 2pq,Comm.Algebra,28(6),2000,2685–2714.

[9]Fang,X.G.,Havas,G.and Praeger,C.E.,On the automorphism groups of quasi-primitive almost simple graphs,J.Algebra,222,1999,271–283.

[10]Fang,X.G.,Ma,X.S.and Wang,J.,On locally primitive Cayley graphs of finite simple groups,J.Combin.Theory Ser.A,118,2011,1039–1051.

[11]Fang,X.G.and Praeger,C.E.,On graphs admitting arc-transitive actions of almost simple groups,J.Algebra,205,1998,37–52.

[12]Fang,X.G.,Praeger,C.E.and Wang,J.,Locally primitive Cayley graphs of finite simple groups,Sci.China Ser.A,44,2001,58–66.

[13]Folkman,J.,Regular line-symmetric graphs,J.Combin Theory Ser.B,3,1967,215–232.

[14]Giudici,M.,Li,C.H.and Praeger,C.E.,Analysing finite locallys-arc transitive graphs,Trans.Amer.Math.Soc.,356,2004,291–317.

[15]Gorenstein,D.,Finite Simple Groups,Plenum Press,New York,1982.

[16]Han,H.and Lu,Z.P.,Semisymmetric graphs of order 6p2and prime valency,Sci.China Math.,55,2012,2579–2592.

[17]Han,H.and Lu,Z.P.,Affine primitive permutation groups and semisymmetric graphs,Electronic J.Combin.,20(2),2013,Research Paper 39.

[18]Han,H.and Lu,Z.P.,Semisymmetric graphs arising from primitive permutation groups of degree 9p,Sci.China Math.,58,2015,to appear.DOI:10.1007/s11425-000-0000-0

[19]Huppert,B.,Endliche Gruppen I,Springer-Verlag,Berlin,1967.

[20]Huppert,B.and Blackburn,N.,Finite Groups II,Springer-Verlag,Berlin,1982.

[21]Iranmanesh,M.A.,On finiteG-locally primitive graphs and the Weiss conjecture,Bull.Austral.Math.Soc.,70,2004,353–356.

[22]Li,C.H.,Lou,B.G.and Pan,J.M.,Finite locally primitive abelian Cayley graphs,Sci.China Math.,54,2011,845–854.

[23]Li,C.H.and Ma,L.,Locally primitive graphs and bidirect products of graphs,J.Aust.Math.Soc.,91,2011,231–242.

[24]Li,C.H.,Pan,J.M.and Ma,L.,Locally primitive graphs of prime-power order,J.Aust.Math.Soc.,86,2009,111–122.

[25]Lu,Z.P.,On the automorphism groups of bi-Cayley graphs,Beijing Daxue Xuebao,39,2003,1–5.

[26]Lu,Z.P.,Wang,C.Q.and Xu,M.Y.,On semisymmetric cubic graphs of order 6p2,Sci.China Ser.A,47(1),2004,1–17.

[27]Malni?,A.,Maru?i?,D.and Wang,C.Q.,Cubic edge-transitive graphs of order 2p3,Discrete Math,274,2004,187–198.

[28]Pan,J.M.,Locally primitive normal Cayley graphs of metacyclic groups,Electron.J.Combin.,16,2009,Research Paper 96.

[29]Praeger,C.E.,Imprimitive symmetric graphs,Ars.Combin.,19A,1985,149–163.

[30]Praeger,C.E.,An o’Nan-Scott theorem for finite quasi-primitive permutation groups and an application to 2-arc transitive graphs,J.London Math.Soc.,47,1992,227–239.

[31]Spiga,P.,OnG-locally primitive graphs of locally twisted wreath type and a conjecture of Weiss,J.Combin.Theory Ser.A,118,2011,2257–2260.

[32]Weiss,R.,s-Transitive graphs,Algebraic methods in graph theory,Colloq.Math.Soc.János Bolyai,25,1981,827–847.