A classification of cubic edge-transitive graphs of order 46p2

Acta Universitatis Apulensis ISSN: 1582-5329 http://www.uab.ro/auajournal/ No. 40/2014 pp. 135-143 doi: 10.17114/j.aua.2014.40.12 A CLASSIFICATION OF CUBIC EDGE-TRANSITIVE GRAPHS OF ORDER 46P 2 M. Alaeiyan, L. Pourmokhtar Abstract. A graph is called edge-transitive, if its automorphisms group acts transitively on the set of its edges. In this paper, we classify all connected cubic edge-transitive graphs of order 46p2 , where p is a prime. 2000 Mathematics Subject Classification: 05C25; 20B25. Keywords: symmetric graphs, semisymmetric graphs, s-regular graphs, regular coverings. 1. Introduction Throughout this paper, graphs are assumed to be finite, simple, undirected and connected. For the group-theoretic concepts and notations not defined here we refer to [19]. For a graph X, we denote by V (X), E(X), A(X) and Aut(X) the vertex set, the edge set, the arc set and the full automorphisms group of X, respectively. For u, v ∈ V (X), denote by {u, v} the edge incident to u and v in X. Let G be a finite group and S a subset of G such that 1 ∈ / S and S = S −1 . The Cayley graph X = Cay(G, S) on G with respect to S is defined to have vertex set V (X) = G and edge set E(X) = {(g, sg)|g ∈ G, s ∈ S}. Clearly, Cay(G, S) is connected if and only if S generates the group G. The automorphism group Aut(X) of X contains the right regular representation GR of G, the acting group of G by right multiplication, as a subgroup, and GR is regular on V (X), that is, GR is transitive on V (X) with trivial vertex stabilizers. A graph X is isomorphic to a Cayley graph on a group G if and only if its automorphism group Aut(X) has a subgroup isomorphic to G, acting regularly on the vertex set. An s-arc in a graph X is an ordered (s + 1)-tuple (v0 , v1 , . . . , vs−1 , vs ) of vertices of X such that vi−1 is adjacent to vi for 1 ≤ i ≤ s and vi−1 6= vi+1 for 1 ≤ i < s. A graph X is said to be s-arc-transitive if Aut(X) acts transitively on the set of its s-arcs. In particular, 0-arc-transitive means vertex-transitive, and 1-arc-transitive 135 M. Alaeiyan, L. Pourmokhtar – A classification of cubic edge-transitive graphs . . . means arc-transitive or symmetric. A graph X is said to be s-regular, if Aut(X) acts regularly on the set of its s-arcs. Tutte [21] showed that every finite connected cubic symmetric graph is s-regular for 1 ≤ s ≤ 5. A subgroup of Aut(X) is said to be s-regular, if it acts regularly on the set of s-arcs of X. If a subgroup G of Aut(X) acts transitively on V (X) and E(X), we say that X is G-vertex-transitive and G-edge-transitive, respectively. In the special case, when G =Aut(X), we say that X is vertex-transitive and edge-transitive, respectively. It can be shown that a G-edge-transitive but not G-vertex-transitive graph X is necessarily bipartite, where the two parts of the bipartition are orbits of G ≤Aut(X). Moreover, if X is regular then these two parts have the same cardinality. A regular G-edge-transitive but not G-vertex-transitive graph will be referred to as a G-semisymmetric graph. In particular, if G =Aut(X) the graph is said to be semisymmetric. The classification of cubic symmetric graphs of different orders is given in many papers. By [3, 4], the cubic s-regular graphs up to order 2048 are classified. Throughout this paper, p and q are prime numbers. The s-regular cubic graphs of some orders such as 2p2 , 4p2 , 6p2 , 10p2 were classified in [9, 10, 11, 12]. Also recently, cubic sregular graphs of order 2pq were classified in [25]. Also, the study of semisymmetric graphs was initiated by Folkman [14]. For example, cubic semisymmetric graphs of orders 6p2 , 8p2 , 4pn and 2pq are classified in [17, 1, 2, 8]. In this paper we classify all cubic edge-transitive (symmetric and also semisymmetric) graphs of order 46p2 as follows. Theorem 1. Let p be a prime. Then the only connected cubic edge-transitive graph of order 46p2 is the 2-regular graph C(N (23, 23, 23)). 2. Preliminaries Let X be a graph and let N be a subgroup of Aut(X). For u, v ∈ V (X), denote by {u, v} the edge incident to u and v in X, and by NX (u) the set of vertices adjacent to u in X. The quotient graph X/N or XN induced by N is defined as the graph such that the set Σ of N -orbits in V (X) is the vertex set of X/N and B, C ∈ Σ are adjacent if and only if there exist u ∈ B and v ∈ C such that {u, v} ∈ E(X). e is called a covering of a graph X with projection ℘ : X e → X if there A graph X e is a surjection ℘ : V (X) → V (X) such that ℘|NXe (ṽ) : NXe (ṽ) → NX (v) is a bijection e of X with a projection for any vertex v ∈ V (X) and ṽ ∈ ℘−1 (v). A covering graph X ℘ is said to be regular (or K-covering) if there is a semiregular subgroup K of the e such that graph X is isomorphic to the quotient graph automorphism group Aut(X) e e → X/K e X/K, say by h, and the quotient map X is the composition ℘h of ℘ and h. 136 M. Alaeiyan, L. Pourmokhtar – A classification of cubic edge-transitive graphs . . . Proposition 1. [15, Theorem 9] Let X be a connected symmetric graph of prime valency and let G be an s-regular subgroup of Aut(X) for some s ≥ 1. If a normal subgroup N of G has more than two orbits, then it is semiregular and G/N is an s-regular subgroup of Aut(XN ), where XN is the quotient graph of X corresponding to the orbits of N . Furthermore, X is a N -regular covering of XN . The next proposition is a special case of [23, Proposition 2.5]. Proposition 2. Let X be a G-semisymmetric cubic graph with bipartition sets U (X) and W (X), where G ≤ A := Aut(X). Moreover, suppose that N is a normal subgroup of G. Then, (1) If N is intransitive on bipartition sets, then N acts semiregularly on both U (X) and W (X), and X is an N -regular covering of a G/N -semisymmetric graph XN . (2) If 3 dose not divide |Aut(X)/N |, then N is semisymmetric on X. Proposition 3. [7, Proposition 2.5] Let X be a connected cubic symmetric graph and G be an s-regular subgroup of Aut(X). Then, the stabilizer Gv of v ∈ V (X) is isomorphic to Z3 , S3 , S3 × Z2 , S4 , or S4 × Z2 for s = 1, 2, 3, 4 or 5, respectively. Proposition 4. [18, Proposition 2.4] The vertex stabilizers of a connected G-semi symmetric cubic graph X have order 2r · 3, where 0 ≤ r ≤ 7. Moreover, if u and v are two adjacent vertices, then the edge stabilizer Gu ∩ Gv is a common Sylow 2-subgroup of Gu and Gv . Now, we have the following obvious fact in the group theory. Proposition 5. Let G be a finite group and let p be a prime. If G has an abelian Sylow p-subgroup, then p does not divide |G′ ∩ Z(G)|. Proposition 6. [24, Proposition 4.4]. Every transitive abelian group G on a set Ω is regular and the centralizer of G in the symmetric group on Ω is G. The next two proposition are the result of [16 , Theorem 1.16]. Proposition 7. Let G be a finite group and let p be a prime, where p | |G| and gcd(m, p) = 1. Therefore, if np (G) ≇ 1(modp2 ), then there are P, R ∈ Sylp (G) such that [P ∩ R : P ] = p and [P ∩ R : R] = p. Proposition 8. Let G be a finite group of order pk n, where k > 0, p is a prime and p ∤ |G|. Moreover, suppose P and R are two distinct Sylow p-subgroups of G such that [P ∩ R : P ] = p. Then [G : NG (P ∩ R)] = n/t, where t ∤ p, t > p. 137 M. Alaeiyan, L. Pourmokhtar – A classification of cubic edge-transitive graphs . . . 3. Main results Let p be an odd prime. Let N (p, p, p) = hxp = y p = z p = 1, [x, y] = z, [z, x] = [z, y] = 1i be a finite group of order p3 and G = ha, b, c, d | a2 = bp = cp = dp = [a, d] = [b, d] = [c, d] = 1, d = [b, c], aba = b−1 , aca = c−1 i be a group of order 2p3 and S = {a, ab, ac}. We write C(N (p, p, p)) = Cay(G, S). By [13, Theorem 3.2], C(N (p, p, p)) is a 2-regular graph of order 2p3 . Let X be a cubic edge-transitive graph of order 46p2 . By [22], every cubic edge and vertex-transitive graph is arc-transitive and consequently, X is either symmetric or semisymmetric. We now consider the symmetric case and then we have the following lemma. Lemma 2. Let p be a prime and let X be a cubic symmetric graph of order 46p2 . Then X is isomorphic to the 2-regular graph C(N (23, 23, 23)). Proof. By [3, 4] there is no symmetric graph of order 46p2 , where p < 7. If p = 23, then by [13, Theorem 3.2], X is isomorphic to the 2-regular graph C(N (23, 23, 23)). To prove the lemma, we only need to show that no cubic symmetric graph of order 46p2 exist, for p ≥ 7, p 6= 23. We suppose to the contrary that X is such a graph. Set A := Aut(X). By Proposition 4, |Av | = 2s−1 · 3, where 1 ≤ s ≤ 5 and hence |A| = 2s · 3 · 23 · p2 . Let N be a minimal normal subgroup of A. Thus, N ∼ = T × T × · · · × T = T k, where T is a simple group. Let N be unsolvable. By Proposition 1 N has at most two orbits on V (X) and hence 23p2 | |N |. Since p ≥ 7, p 6= 23 and 32 ∤ |A|, one has k = 1 and hence N ∼ = T . So |N | = 2t .23.p2 or 2t .3.23.p2 , where 1 ≤ t ≤ s. Let q be a prime .Then by [6], a non-abelian simple {2, p, q}-group is one of the following groups A5 , A6 , P SL(2, 7), P SL(2, 8), P SL(2, 17), P SL(3, 3), P SU (3, 3), P SU (4, 2) (1) With orders 22 .3.5, 23 .32 .5, 23 .3.7, 23 .32 .7, 24 .32 .17, 24 .33 .13, 25 .33 .7, 26 .34 .5, respectively. This implies that for p ≥ 7, there is no simple group of order 2t .23.p2 . Hence |N | = 2t .3.23.p2 . Assume that L is a proper subgroup of N . If L is unsolvable, then L has a non-abelian simple composite factor L1 /L2 . Since p ≥ 11 and |L1 /L2 ||2t .3.23.p2 , by simple group listed in 1, L1 /L2 cannot be a {2, 3, 23}−, {2, 3, p} − or{2, 23, p}group. Thus, L1 /L2 is a {2, 3, 23, p}-group. One may assume that |L| = 2r .3.23.p2 or 2r .3.23.p, where r ≥ 2. Let |L| = 2r .3.23.p2 . Then |N : L| ≤ 8 because |N | = 2t .3.23.p2 . Consider the action of N on the right cosets of by right multiplication, and the simplicity of N implies that this action is faithful. It follows N ≤ S8 and hence p ≤ 7. Since p ≥ 7, one has p = 7 and hence N = 2t .3.23.72 . But by [6], there is no non-abelian simple group of order 2t .3.23.72 , a contradiction. Thus, L is 138 M. Alaeiyan, L. Pourmokhtar – A classification of cubic edge-transitive graphs . . . solvable and hence N is a minimal non-abelian simple group, that is, N is a nonabelian simple group and every proper subgroup of N is solvable. By [20, Corollary 1], N is one of the groups in Table I. It can be easily verified that the order of groups in Table I are not of the form 2r .3.23.p2 . Thus |L| = 2r .3.23.p. By the same argument as in the preceding paragraph (replacing N by L) L is one of the groups in Table I. Since |L| = 2r .3.23.p, the possible candidates for L is P SL(2, m). Clearly, m = p. We show that |L| < 1025 . If 23 ∤ (p − 1)/2, then (p − 1)/2|96, which implies that p ≤ 193. If p = 193, then 26 ||L|, a contradiction. Thus p < 193 and hence p ≤ 97 because (p − 1)/2|96. It follows that |L| ≤ 96.23.97 = 214176. If 23|(p − 1)/2, Then p + 1|96. Consequently p ≤ 47, implying |L| ≤ 96.23.47 < 214176. Thus, |L| ≤ 214176. Then by [6], is isomorphic to P SL(2, 23) or P SL(2, 47). It follows that p = 11 or 47 and hence |N | = 2t .3.23.112 or 2t .3.23.472 , which is impossible by [6]. Table I. The possible for non-abelian simple group N N P SL(2, m), m > 3 a prime and m2 6= 3 (mod p2 ) P SL(2, 2n ), n a prime P SL(2, 3n ), n an odd prime P SL(3, 3), n a prime Suzuki group Sz(2n ), n an odd prime |N| − 1)(m + 1) 2n (22n − 1) 1 n 2n 3 (3 − 1) 2 1 3 4 .3 .2 3 2n 2n 2 (2 + 1)(2n − 1) 1 m(m 2 Hence, N is solvable and so elementary abelian. Again by Proposition 1, N is semiregular, implying |N | | 46p2 . Consequently, N ∼ = Z2 , Zp × Zp , Zp or Z23 . If N ∼ = Z2 , then by Proposition 1, XN is a cubic graph of odd order 23p2 , a contradiction. Also, if N ∼ = Zp × Zp , then by Proposition 1, XN is a cubic symmetric graph of order 46. But, by [3, 4] there is no symmetric cubic graph of order 46, a contradiction. Suppose now that N ∼ = Zp . Set C := CA (N ) the centralizer of N in A. Let K be a Sylow p-subgroup of A. Since K is an abelian group and N < K, p2 | |C|. Suppose that C ′ is the derived subgroup of C. This forces p2 ∤ |C ′ | and hence C ′ has more than two orbits on V (X). By Proposition 1, C ′ is semiregular and consequently |C ′ | | 46p2 . Since C/C ′ is an abelian group and p2 ∤ |C ′ |, then C/C ′ has a normal Sylow p-subgroup, say H/C ′ , which is normal in A/C ′ . Thus H ⊳ A and p2 | |H|. Also |H| | 46p2 because |C ′ | | 46p2 and |H/C ′ | | p2 . Hence H has a characteristic Sylow p-subgroup of order p2 , say K, which is normal in A. Then by Proposition 1, XK is a cubic symmetric graph of order 46, a contradiction. Now, suppose that N ∼ = Z23 . Since N has more than two orbits, then by Proposition 1, N is semiregular and the quotient XN is a cubic A/N -symmetric graph of order 2p2 and A/N is an arc-transitive subgroup of Aut(XN ). Suppose first that 139 M. Alaeiyan, L. Pourmokhtar – A classification of cubic edge-transitive graphs . . . p = 7 and T /N be a minimal normal subgroup of A/N . Thus by [11, Lemma 3.1], T /N is 7-subgroup abelian elementary. So |T /N | = 7 or 72 . Consequently |T | = 23.7 or 23.72 . It is easy to see that the Sylow 7-subgroup of T is normal in A, and by the same argument as the previous paragraph, we get a similar contradiction. We suppose now p = 11 and let M/N be the Sylow p-subgroup of A/N . Then, M/N by [10, Lemma 3.1], is normal in A/N . It follows that M is normal in A and |M/N | = 112 . It implies that |M | = 23.112 . Let n11 be the number of the Sylow 11-subgroups of M . Thus n11 | 23. So n11 = 1 or 23. If n11 = 1, then the Sylow 11-subgroup of M is normal in A, so we get a contradiction. Also, if n11 = 23, then by Proposition 7, M has two distinct Sylow 11-subgroups, say P and R, such that [P ∩ R : P ] = 11 and [P ∩ R : R] = 11. Let NM (P ∩ R) be normalizer P ∩ R in M . According to Proposition 8 , [M : NM (P ∩ R] = 1 and hence P ∩ R is normal in M . Since M is characteristic in A, so P ∩ R is normal in A. Again A has a normal subgroup of order p(= 11), a contradiction. We now suppose that p ≥ 13, p 6= 23. Then [11, Theorem 3.2], the Sylow psubgroup of Aut(XN ) is normal. Consequently, the Sylow p-subgroup of A/N , say M/N , is normal. Thus, M is normal in A and |M | = 23p2 . It follows that the Sylow p-subgroup of A, say K, is normal. Then by Proposition 1, XK is a cubic symmetric graph of order 46, a contradiction. Hence, the result now follows. Now, we study the semisymmetric case, and we have the following lemma. Lemma 3. Let p be a prime. Then, there is no cubic semisymmetric graph of order 46p2 . Proof. Let X be a cubic semisymmetric graph of order 46p2 . Denote by U (X) and W (X) the bipartition sets of X, where |U (X)| = |W (X)| = 23p2 . For p = 2, 3, by [5] there is no cubic semisymmetric graph of order 46p2 . Thus we can assume that p ≥ 5. Set A := Aut(X) and let Q := Op (A) be the maximal normal psubgroup of A. By Proposition 4, we have |Av | = 2r · 3, where 0 ≤ r ≤ 7 and hence |A| = 2r · 3 · 23 · p2 . Let N be a minimal normal subgroup of A. If N is unsolvable, then N × T × = T k , where T is a non-abelian {2, 3, 23} or {2, 3, 23, p}simple group. By [6], T ∼ = A5 , P SL(2, 7), P SL(2, 23) or P SL(2, 47) with orders 22 · 3 · 7, 23 · 3 · 7, 23 · 3 · 11 · 23 and 24 · 3 · 23 · 47, respectively. But 32 ∤ |N | and hence k = 1. So N ∼ = T . Since 3 ∤ |A/N |, by Proposition 3, N must be semisymmetric on X and then 23p2 | |N |, a contradiction. So N is solvable and so elementary abelian. Thus N acts intransitively on U (X) and W (X) and by Proposition 2, it is semiregular on each partition. Hence |N | | 23p2 . So |N | = 23, p or p2 . We show that |Q| = p2 as follows. First Suppose that Q = 1. It implies that N ∼ = Z23 . Let XN be the quotient graph of X relative to N , where XN is a cubic A/N -semisymmetric graph of order 140 M. Alaeiyan, L. Pourmokhtar – A classification of cubic edge-transitive graphs . . . 2p2 . By [11], XN is a vertex-transitive graph. So XN is a cubic symmetric graph of order 2p2 . Suppose that T /N be a minimal normal subgroup in A/N . First suppose that p = 5, by [11, Lemma 3.1], T /N is 5-subgroup abelian elementary. So |T /N | = 5 or 52 and hence |T | = 23 · 5 or 23 · 52 . It follows the Sylow 5-subgroup T is normal in A. This is a contrary with |Q| = 1. Now, suppose p = 7, 11. Then, by similar argument as above, we get a contradiction. Therefore, we can suppose that p ≥ 13. By [11, Lemma 3.1], Sylow p-subgroup of A/N is normal, say M/N . So |M/N | = p2 and hence |M | = 23p2 . Clearly, the Sylow p-subgroup M is normal in A, a contradiction. We now suppose that |Q| = p. Since |N | | 23p2 , then we have two cases: N ∼ = Z23 and N ∼ Z . = p Case I. N ∼ = Z23 . By Proposition 2, XN is a cubic A/N -semisymmetric graph of order 2p2 . Let T /N be a minimal normal subgroup of A/N . If T /N is an unsolvable group, then by [6], T /N ∼ = P SL(2, 7). Thus |T | = 23 · 3 · 23 · 7. Since 3 ∤ |A/T |, then by Proposition 2, T is semisymmetric on X. Consequently 72 | |T |, a contradiction. Hence T /N is solvable and so elementary abelian. If |T /N | = p2 , then |T | = 23p2 . By a similar way as above, we get, the Sylow p-subgroup of T is characteristic and consequently normal in A. It contradicts our assumption that |Q| = p. Therefore T /N intransitively on bipartition sets of XN and by Proposition 2, it is semiregular on each partition, which force |T /N | | p2 . Hence |T /N | = p and so |T | = 23p. Since T acts intransitively on bipartition sets of X, by Proposition 2, XT is a cubic A/T -semisymmetric graph of order 2p. Let K/T be a minimal normal subgroup of A/T . Clearly N ⊳ K. If K/N is unsolvable then by [6], K/N ∼ = P SL(2, 7) and so 3 |K| = 2 · 3 · 23 · 7. Since K ⊳ A and 3 dose not divide |A/K|, then by Proposition 2, K is semisymmetric on X. Therefore 23 · 72 | |K|, a contradiction. It follows that K/N is solvable and since N is solvable, K is solvable. Consequently K/T is solvable and so elementary abelian. If K/T acts transitively on any partition of XT , then by Proposition 6, K/T is regular and hence |K/T | = p. Therefore, |K| = 23p2 . Similarly as the case |Q| = 1, in this case, we get that p 6= 5, 7, 11 and the Sylow psubgroup K is characteristic and so normal in A, a contrary to this fact that |Q| = p. Thus K/T acts intransitively on each partition of XT and by Proposition 2, K/T is semiregular on two partitions. It implies that |K/T | = p and so |K| = 23p2 , a similar contradiction is obtained. Case II. N ∼ = Zp . By Proposition 2, XN is a cubic A/N -semisymmetric graph of order 46p. Let T /N be a minimal normal subgroup of A/N . By a similar way as above, T /N is solvable and so elementary abelian. By Proposition 2, T /N is semiregular. It implies that |T /N | | 23p. If |T /N | = p, then |T | = p2 , a contrary to this fact that |Q| = p. Hence |T /N | = 23 and so |T | = 23p. By Proposition 2, XT 141 M. Alaeiyan, L. Pourmokhtar – A classification of cubic edge-transitive graphs . . . is a cubic A/T -semisymmetric graph of order 2p. Thus by a similar way as case I, we get a contradiction. Therefore |Q| = p2 and so by Proposition 2, X is a regular Q-covering of an A/Q-semisymmetric graph of order 46. But it is impossible because by [4, 5] there is no edge-transitive graph of order 46. The result now follows. Proof of Theorem Now we complete the proof of the main theorem. Let X is a connected cubic edge-transitive graph of order 46p2 , where p is a prime. We know that every cubic edge-transitive graph is either symmetric or semisymmetric. Therefore, by Lemmas 2 and 3 the proof is completed. References [1] M. Alaeiyan, M. Ghasemi, Cubic edge-transitive graphs of oredr 8p2 , Bull. Aust. Math. Soc. 77, 2 (2008), 315-324. [2] M. Alaeiyan, B. N. Onagh, Semisymmetric cubic graphs of order 4pn , Acta Univ. Apulensis Math. Inform. 19, (2009), 153-158. [3] M. Conder, Trivalent (cubic) symmetric graphs on up to 2048 vertices, (2006). http://www.math.auckland.ac.nz/∼conder/∼conder/symmcubic2048list.txt. [4] M. Conder, R. Nedela, A refined classification of cubic symmetric graphs, J. Algebra. 322, 3 (2009), 722-740. [5] M. Conder, A. Malnič, D. Marušič, P. Potočnik, A census of semisymmetric cubic graphs on up to 768 vertices, J. Algebr. Comb. 23, 3 (2006), 255-294. [6] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, An Atlas of Finite Groups, Oxford University Press, Oxford, (1985). [7] D. Ž. Djoković, G. L. Miller, Regular groups of automorphisms of cubic graphs, J. Combin. Theory Ser. B. 29, 2 (1980), 195-230. [8] S. Du, M. Xu, A classification of semisymmetric graphs of order 2pq, Com. Algebra. 28, 6 (2000), 2685-2715. [9] Y. Q. Feng, J. H. Kwak, One regular cubic graphs of order a small number times a prime or a prime square, J. Aust. Math. Soc. 76, 03 (2004), 345-356. [10] Y. Q. Feng, J. H. Kwak, Classifying cubic symmetric graphs of order 10p or 10p2 , Sci. China Ser. A. Math. 49, 3 (2006), 300-319. [11] Y. Q. Feng, J. H. Kwak, Cubic symmetric graphs of order twice an odd primepower, J. Aust. Math. Soc. 81, 02 (2006), 153-164. [12] Y. Q. Feng, J. H. Kwak, Cubic symmetric graphs of order a small number times a prime or a prime square, J. Combin. Theory Ser. B. 97, 4 (2007), 627-646. [13] Y. Q. Feng, J. H. Kwak, M. Y. Xu, Cubic s-regular graphs of order 2p3 , J. Graph Theory. 52, 4 (2006), 341-352. 142 M. Alaeiyan, L. Pourmokhtar – A classification of cubic edge-transitive graphs . . . [14] J. Folkman, Regular line-symmetric graphs, J. Combin. Theory. 3, 3 (1967), 215-232. [15] J. L. Gross, T. W. Tucker, Generating all graph coverings by permutation voltages assignments, Discrete Math. 18, 3 (1977), 273-283. [16] I. M. Isaacs, Finite Group Theory, American Mathematical Society Providence, Rhode Island, (2008). [17] Z. Lu, C.Q. Wang, M.Y. Xu, On semisymmetric cubic graphs of order 6p2 , Sci. China Math. 47, 1 (2004), 1-17. [18] A. Malnič, D. Marušič, C. Wang, Cubic edge-transitive graphs of order 2p3 , Discrete Math. 274, 1 (2004), 187-198. [19] J. S. Rose, A Course On Group Theory, Courier Dover Publications, (1978). [20] J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable I, Bull. amer. Math. Soc. 74, 3 (1968), 383-437. [21] W. T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43, 04 (1947) 459-474. [22] W. T. Tutte, Connectivity in graphs, University of Toronto Press, (1966). [23] C. Q. Wang, T. S. Chen, Semisymmetric cubic graphs as regular covers of K3,3 , Acta Math. Sin. 24, 3 (2008), 405-416. [24] H. Wielandant, Finite Permutation Groups, Acadamic Press. New York, (1964). [25] J. X. Zhou, Y. Q. Feng, Cubic vertex-transitive graphs of order 2pq, J. Graph Theory. 65, 4 (2010), 285-302. Mehdi Alaeiyan (corresponding author) Department of Mathematics, Faculty of Science, Iran University of Science and Technology Tehran, Iran. email: [email protected] Laleh Pourmokhtar Department of Mathematics, Faculty of Science, Iran University of Science and Technology Tehran, Iran. email: laleh [email protected] 143