Home Mathematics On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
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On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq

  • Bo Ling EMAIL logo , Ting Lan and Suyun Ding
Published/Copyright: December 31, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

A graph is said to be symmetric if its automorphism group is transitive on its arcs. This article is one of a series of articles devoted to characterizing prime-valent arc-transitive graphs of square-free order or twice square-free order. In this article, we determine all 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq.

Keywords: symmetric graph; normal quotient; automorphism group
MSC 2010: 05C25; 05E18

1 Introduction

For a simple, connected, and undirected graph Γ , the vertex set and arc set of Γ are denoted by V Γ and A Γ , respectively. Let G be a subgroup of the full automorphism group Aut Γ of Γ . Then, Γ is called G-vertex-transitive and G-arc-transitive if G is transitive on V Γ and A Γ , respectively. An arc-transitive graph is also called symmetric. It is well known that Γ is G -arc-transitive if and only if G is transitive on V Γ and the stabilizer G α ≔ { g ∈ G ∣ α g = α } for some α ∈ V Γ is transitive on the neighbor set Γ ( α ) of α in Γ .

For a group G and a subset S = S − 1 ≔ { s − 1 ∣ s ∈ S } of G , the Cayley graph Cay ( G , S ) is a graph with vertex set G and edge set { { g , s g } ∣ g ∈ G , s ∈ S } . It is well known that the right multiplication of G , say R ( G ) , and the set Aut ( G , S ) ≔ { α ∈ Aut ( G ) ∣ S α = S } are groups of automorphisms of Cay ( G , S ) . The Cayley graph Cay ( G , S ) is called normal if the right multiplication of G is normal in Aut ( Cay ( G , S ) ) . The following Cayley graphs of dihedral groups are denoted by CD 2 p q k .

Example 1.1

Let G = ⟨ a , b ∣ a p q = b 2 = 1 , a b = a − 1 ⟩ ≅ D 2 p q , and let k be a solution of the equation

x 6 + x 5 + ⋯ + x + 1 ≡ 0 ( mod p q ) .

Set

CD 2 p q k = Cay ( G , { b , a b , a k + 1 b , … , a k 5 + k 4 + ⋯ + k + 1 b } ) .

The study of graphs with square-free order has a long history, see, e.g., [1,2, 3,4]. In recent work [5], the authors gave a characterization for connected prime-valent arc-transitive graphs of square-free order. This article is devoted to classifying 7-valent arc-transitive graphs of order 2pq, which gives supplementary proof of Lemma 2.9 in [5, Lemma 2.9]. The first result of this article is the following theorem.

Theorem 1.2

Let Γ be a 7-valent symmetric graph of order 2pq, where q > p ≥ 3 are primes. Then one of the following statements holds:

  1. Γ ≅ CD 2 p q k and Aut Γ ≅ D 2 p q : Z 7 , where p ∣ q − 1 . Up to isomorphism, there is only one such graph for p = 7 and there are exactly six such graphs for p > 7 .

  2. Γ lies in Table 1.

Table 1

Connected 7-valent symmetric graphs of order 2pq

Row Γ ( p , q ) Aut Γ ( Aut Γ ) α Transitivity Remark
1 C 78 1 (3,13) PGL(2,13) D 28 1-transitive No bipartite
2 C 78 2 (3,13) PSL(2,13) D 14 1-transitive No bipartite
3 C 310 (5,31) PSL ( 5 , 2 ) . Z 2 Z 2 6 : ( SL ( 2 , 2 ) × SL ( 3 , 2 ) ) 3-transitive Bipartite
4 C 30 (3,5) S 8 Z 2 3 : SL ( 3 , 2 ) 2-transitive Bipartite

The method used in this article for classifying symmetric graphs of square-free order is also applicable to classifying symmetric graphs of twice square-free order. See the following two theorems.

Theorem 1.3

Let Γ be a connected symmetric graph of order 4 p with valency 11, where p is a prime, then p = 3 and Γ = K 12 , the complete graph of order 12.

Theorem 1.4

Let Γ be a connected symmetric graph of order 4pq with valency 11, where p > q ≥ 3 are distinct primes, then Γ ≅ G 60 , G 532 , or G 276 i for 1 ≤ i ≤ 4 , with their automorphism groups Aut Γ and vertex stabilizers ( Aut Γ ) α listed in Table 2, where α is a vertex.

Table 2

Connected 11-valent symmetric graphs of order 4pq

Graph Aut Γ ( Aut Γ ) α ( q , p ) Bipartite?
G 60 PGL(2,11) F 22 (3,5) No
G 276 1 PGL(2,23) D 44 (3, 23) Yes
G 276 i , 2 ≤ i ≤ 4 PSL(2,23) D_22 (3, 23) No
G 532 J 1 × Z 2 PSL(2,11) (7, 19) Yes

2 Preliminaries

We now give some necessary preliminary results. The first one is a property of the Fitting subgroup, see [6, P. 30, Corollary].

Lemma 2.1

Let F be the Fitting subgroup of a group G. If G is soluble, then F ≠ 1 and the centralizer C G ( F ) ≤ F .

We shall need information of maximal subgroups of PSL ( 2 , r ) and PGL ( 2 , r ) , where r is an odd prime, refer to [7, Section 239] and [8, Theorem 2].

Lemma 2.2

Let G = PSL ( 2 , r ) or PGL ( 2 , r ) and let M be a maximal subgroup of G, where r ≥ 5 is a prime.

  1. If G = PSL ( 2 , r ) , then M ∈ { D r − 1 , D r + 1 , Z r : Z ( r − 1 ) / 2 , A 4 , S 4 , A 5 } .

  2. If G = PGL ( 2 , r ) , then M ∈ { D 2 ( r − 1 ) , D 2 ( r + 1 ) , Z r : Z r − 1 , S 4 , PSL ( 2 , r ) } .

By [9], we have the next lemma.

Lemma 2.3

Let Γ = Cay ( G , S ) be a normal Cayley graph on G. Then, ( Aut Γ ) 1 = Aut ( G , S ) , where 1 is the identity of G.

For a graph Γ and a positive integer s , an s -arc of Γ is a sequence α 0 , α 1 , … , α s of vertices such that α i − 1 and α i are adjacent for 1 ≤ i ≤ s and α i − 1 ≠ α i + 1 for 1 ≤ i ≤ s − 1 . In particular, a 1-arc is just an arc. Then, Γ is called ( G , s ) -arc-transitive with G ≤ Aut Γ if G is transitive on the set of s -arcs of Γ . A ( G , s ) -arc-transitive graph is called ( G , s ) -transitive if it is not ( G , s + 1 ) -arc-transitive. In particular, a graph Γ is simply called s-transitive if it is ( Aut Γ , s ) -transitive.

The following lemma is about the stabilizers of arc-transitive 7-valent graphs, refer to [10, Corollary 2.2] and [11, Theorem 3.4].

Lemma 2.4

Let Γ be a 7-valent ( G , s ) -transitive graph, where G ≤ Aut Γ and s ≥ 1 . Let α ∈ V Γ . Then, one of the following statements holds:

  1. If G α is soluble, then s ≤ 3 and ∣ G α ∣ ∣ 252 . Furthermore, the couple ( s , G α ) lies in the following table.

    s 1 2 3
    G α Z 7 , F 14 , F 21 , F 14 × Z 2 , F 21 × Z 3 F 42 , F 42 × Z 2 , F 42 × Z 3 F 42 × Z 6

  2. If G α is insoluble, then ∣ G α ∣ ∣ 2 24 ⋅ 3 4 ⋅ 5 2 ⋅ 7 .

From [12, pp. 134–136], we can obtain the following two lemmas by checking the orders of nonabelian simple groups. The arguments in the proof of Lemmas 2.5 and 2.6 are heavily relying on the classification of finite simple groups.

Lemma 2.5

Let q > p ≥ 3 be primes, and let T be a nonabelian simple group of order 2 i ⋅ 3 j ⋅ 5 k ⋅ 7 ⋅ p ⋅ q , where 2 ≤ i ≤ 25 , 0 ≤ j ≤ 4 , and 0 ≤ k ≤ 2 . Then, T is listed in Table 3.

Table 3

Simple group T with order dividing 2 25 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ p ⋅ q

T ∣ T ∣ ( p , q ) T ∣ T ∣ ( p , q )
M 22 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 ( 3 , 11 ) , ( 5 , 11 ) PSL ( 2 , 2 6 ) 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 13 ( 3 , 13 ) , ( 5 , 13 )
M 23 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 23 ( 11 , 23 ) PSL ( 2 , 2 9 ) 2 9 ⋅ 3 3 ⋅ 7 ⋅ 19 ⋅ 73 ( 19 , 73 )
M 24 2 10 ⋅ 3 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 23 ( 11 , 23 ) PSL ( 2 , 27 ) 2 2 ⋅ 3 3 ⋅ 7 ⋅ 11 ( 3 , 11 )
J 1 2 3 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 19 ( 11 , 19 ) PSL ( 2 , 125 ) 2 2 ⋅ 3 2 ⋅ 5 3 ⋅ 7 ⋅ 31 ( 5 , 31 )
J 2 2 7 ⋅ 3 3 ⋅ 5 2 ⋅ 7 ( 3 , 5 ) PSL ( 2 , 49 ) 2 4 ⋅ 3 ⋅ 5 2 ⋅ 7 2 ( 3 , 7 ) , ( 5 , 7 )
HS 2 9 ⋅ 3 2 ⋅ 5 3 ⋅ 7 ⋅ 11 ( 5 , 11 ) PSU ( 3 , 5 ) 2 4 ⋅ 3 2 ⋅ 5 3 ⋅ 7 ( 3 , 5 )
A 7 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 ( 3 , 5 ) PSL ( 3 , 8 ) 2 9 ⋅ 3 4 ⋅ 7 ⋅ 19 ( 3 , 19 )
A 8 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 ( 3 , 5 ) D 4 ( 2 ) 2 12 ⋅ 3 5 ⋅ 5 2 ⋅ 7 ( 3 , 5 )
A 9 2 6 ⋅ 3 4 ⋅ 5 ⋅ 7 ( 3 , 5 ) D 4 3 ( 2 ) 2 12 ⋅ 3 4 ⋅ 7 2 ⋅ 13 ( 7 , 13 )
A 10 2 7 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ( 3 , 5 ) PSp ( 8 , 2 ) 2 16 ⋅ 3 5 ⋅ 5 2 ⋅ 7 ( 3 , 5 )
A 11 2 7 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ 11 ( 3 , 11 ) , ( 5 , 11 ) PSL ( 4 , 4 ) 2 12 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ 17 ( 3 , 17 ) , ( 5 , 17 )
A 12 2 9 ⋅ 3 5 ⋅ 5 2 ⋅ 7 ⋅ 11 ( 3 , 11 ) PSL ( 5 , 2 ) 2 10 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 31 ( 3 , 31 ) , ( 5 , 31 )
Sz ( 8 ) 2 6 ⋅ 5 ⋅ 7 ⋅ 13 ( 5 , 13 ) PSp ( 4 , 8 ) 2 12 ⋅ 3 4 ⋅ 5 ⋅ 7 2 ⋅ 13 ( 7 , 13 )
PSU ( 3 , 8 ) 2 9 ⋅ 3 4 ⋅ 7 ⋅ 19 ( 3 , 19 ) D 4 2 ( 2 ) 2 12 ⋅ 3 4 ⋅ 5 ⋅ 7 ⋅ 17 ( 3 , 17 ) , ( 5 , 17 )
PSp ( 6 , 2 ) 2 9 ⋅ 3 4 ⋅ 5 ⋅ 7 ( 3 , 5 ) G 2 ( 4 ) 2 12 ⋅ 3 3 ⋅ 5 2 ⋅ 7 ⋅ 13 ( 3 , 13 ) , ( 5 , 13 )
PSL ( 4 , 2 ) 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 ( 3 , 5 ) PSL ( 3 , 16 ) 2 12 ⋅ 3 2 ⋅ 5 2 ⋅ 7 ⋅ 13 ⋅ 17 ( 13 , 17 )
PSL ( 3 , 4 ) 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 ( 3 , 5 ) PSL ( 2 , q ) q ( q + 1 ) ( q − 1 ) 2

Proof

If T is a sporadic simple group, by [12, pp. 135–136], T = M 22 , M 23 , M 24 , J 1 , J 2 , or HS . If T = A n is an alternating group, since 2 10 does not divide ∣ T ∣ , we have n ≤ 13 , it then follows that T = A 7 , A 8 , A 9 , A 10 , A 11 , or A 12 in Table 3.

Suppose now T = X ( q ) is a simple group of Lie type, where X is one type of Lie groups, and q = p f is a prime power. If p ≥ 3 , as ∣ T ∣ contains at most five 3-factors, three 5-factors, and two 7-factors, it easily follows from [12, p. 135] that the only possibility is T = PSL ( 2 , q ) , PSL ( 2 , 27 ) , PSL ( 2 , 125 ) , PSL ( 2 , 49 ) , or PSU ( 3 , 5 ) . Similarly, if p = 2 , then we have T = Sz ( 8 ) , PSU ( 3 , 8 ) , PSp ( 6 , 2 ) , PSL ( 4 , 2 ) , PSL ( 3 , 4 ) , PSL ( 3 , 8 ) , PSL ( 2 , 2 6 ) , PSL ( 2 , 2 9 ) , D 4 ( 2 ) , D 4 3 ( 2 ) , PSp ( 8 , 2 ) , PSL ( 4 , 4 ) , PSL ( 5 , 2 ) , PSp ( 4 , 8 ) , D 4 2 ( 2 ) , G 2 ( 4 ) , or PSL ( 3 , 16 ) .□

Lemma 2.6

Let T be a nonabelian simple group and let p > q be two distinct odd primes. Suppose that 11 ∣ ∣ T ∣ and ∣ T ∣ ∣ 2 18 ⋅ 3 8 ⋅ 5 4 ⋅ 7 2 ⋅ 11 ⋅ p ⋅ q , then T lies in Table 4.

Table 4

Simple group T with order dividing 2 18 ⋅ 3 8 ⋅ 5 4 ⋅ 7 2 ⋅ 11 ⋅ p ⋅ q

Part T ∣ T ∣ T ∣ T ∣
1 M 11 2 4 ⋅ 3 2 ⋅ 5 ⋅ 11 M 12 2 6 ⋅ 3 3 ⋅ 5 ⋅ 11
M 22 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 M 23 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 23
M 24 2 10 ⋅ 3 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 23 J 1 2 3 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 19
HS 2 9 ⋅ 3 2 ⋅ 5 3 ⋅ 7 ⋅ 11 M c L 2 7 ⋅ 3 6 ⋅ 5 3 ⋅ 7 ⋅ 11
Suz 2 13 ⋅ 3 7 ⋅ 5 2 ⋅ 7 ⋅ 11 ⋅ 13 C o 3 2 10 ⋅ 3 7 ⋅ 5 3 ⋅ 7 ⋅ 11 ⋅ 23
C o 2 2 18 ⋅ 3 6 ⋅ 5 3 ⋅ 7 ⋅ 11 ⋅ 23 F i 22 2 17 ⋅ 3 9 ⋅ 5 2 ⋅ 7 ⋅ 11 ⋅ 13
2 A 11 2 7 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ 11 A 12 2 9 ⋅ 3 5 ⋅ 5 2 ⋅ 7 ⋅ 11
A 13 2 9 ⋅ 3 5 ⋅ 5 2 ⋅ 7 2 ⋅ 11 ⋅ 13 A 14 2 10 ⋅ 3 5 ⋅ 5 2 ⋅ 7 2 ⋅ 11 ⋅ 13
A 15 2 10 ⋅ 3 6 ⋅ 5 3 ⋅ 7 2 ⋅ 11 ⋅ 13 A 16 2 10 ⋅ 3 6 ⋅ 5 3 ⋅ 7 2 ⋅ 11 ⋅ 13
A 17 2 14 ⋅ 3 6 ⋅ 5 3 ⋅ 7 2 ⋅ 11 ⋅ 13 ⋅ 17 A 18 2 15 ⋅ 3 8 ⋅ 5 3 ⋅ 7 2 ⋅ 11 ⋅ 13 ⋅ 17
3 PSL(2,11) 2 2 ⋅ 3 ⋅ 5 ⋅ 11 PSL( 2 , 1 1 2 ) 2 3 ⋅ 3 ⋅ 5 ⋅ 1 1 2 ⋅ 61
PSL( 2 , 2 5 ) 2 5 ⋅ 3 ⋅ 11 ⋅ 31 PSL( 2 , 2 10 ) 2 10 ⋅ 3 ⋅ 5 2 ⋅ 11 ⋅ 31 ⋅ 41
PSL( 2 , 3 5 ) 2 2 ⋅ 3 5 ⋅ 1 1 2 ⋅ 61 PSU 5 (2) 2 10 ⋅ 3 5 ⋅ 5 ⋅ 11
PSU 6 (2) 2 15 ⋅ 3 6 ⋅ 5 ⋅ 7 ⋅ 11 PSL( 2 , p ) p ( p − 1 ) ( p + 1 ) / 2

Proof

Assume T is a sporadic simple group. Then, by checking the order of sporadic simple group in [12], we have Part 1 of the table.

Assume T = A n is an alternating group with n ≥ 5 . Since 11 ∣ ∣ T ∣ , n ≥ 11 ; and since ∣ T ∣ has at most seven distinct prime divisors, n ≤ 18 . Then, we have Part 2 of the table.

Assume from now on that T is a simple group of Lie type over a field G F ( r ) of order r = t e , where t is a prime. Note that the order of T is not divisible by 2 19 , 3 10 , 5 6 , 7 4 , 1 1 3 , and s 2 , where s > 11 is a prime.

Assume first that T is a simple exceptional group. By [12], we can easily rule out F 4 ( r ) , E 6 ( r ) , E 6 2 ( r ) , E 7 ( r ) , and E 8 ( r ) as r 19 ∣ ∣ T ∣ if T is one of them. Since 11 ∤ ∣ F 4 2 ( 2 ) ∣ , T ≠ F 4 2 ( 2 ) . If T = F 4 2 ( r ) with r = 2 2 m + 1 ≥ 2 3 , then r 12 ∣ ∣ T ∣ , and hence 2 36 ∣ ∣ T ∣ , a contradiction. If T = D 4 3 ( r ) , then r 12 ∣ ∣ T ∣ , and hence T = D 4 3 ( 2 ) . However 11 ∤ ∣ D 4 3 ( 2 ) ∣ , a contradiction. If T = G 2 ( r ) , then r 6 ∣ ∣ T ∣ , and hence the possibilities are G 2 ( 2 ) , G 2 ( 4 ) , G 2 ( 8 ) , and G 2 ( 3 ) . However, a computation shows that 11 does not divide the orders of these four groups, a contradiction. If T = B 2 2 ( r ) with r = 2 2 m + 1 ≥ 2 3 (noting that B 2 2 ( 2 ) is solvable), then r 2 ∣ ∣ T ∣ , and hence the possibilities are B 2 2 ( 2 3 ) , B 2 2 ( 2 5 ) , B 2 2 ( 2 7 ) , and B 2 2 ( 2 9 ) . However 11 does not divide the orders of these four groups, a contradiction. If T = G 2 2 ( r ) with r = 3 2 m + 1 ≥ 3 3 (noting that G 2 2 ( 3 ) ≅ PSL ( 2 , 8 ) ⋅ 3 is not a simple group, and 11 ∤ ∣ PSL ( 2 , 8 ) ∣ ), then r 3 ∣ ∣ T ∣ , and hence 3 9 ∣ ∣ T ∣ . Then, T = G 2 2 ( 3 3 ) . However, 11 ∤ ∣ G 2 2 ( 3 3 ) ∣ , a contradiction. To summary, we have shown that T is not a simple exceptional group.

Assume next that T is a classical group. Note that r n ( n − 1 ) / 2 ∣ ∣ P G L ( n , r ) ∣ and ∣ PSU n ( r ) ∣ , r m 2 ∣ ∣ PS p 2 m ( r ) ∣ and ∣ P Ω 2 m + 1 ( r ) ∣ , and r m ( m − 1 ) / 2 ∣ ∣ P Ω 2 m ± ( r ) ∣ . Considering the isomorphisms between classical groups (see [12]), the possibilities of T are as follows:

PSL ( 2 , r ) with r divides one of { 2 18 , 3 9 , 5 5 , 7 3 , 1 1 2 , p ( p > 11 ) } , PSL 3 ( 2 k ) for 1 ≤ k ≤ 6 , PSL 3 ( 3 ) , PSL 3 ( 3 2 ) , PSL 3 ( 3 3 ) , PSL 3 ( 5 ) , PSL 3 ( 7 ) , PSL 4 ( 2 k ) for 1 ≤ k ≤ 3 , PSL 4 ( 3 ) , PSL 5 ( 2 ) , PSL 6 ( 2 ) , PSU 3 ( 2 k ) for 2 ≤ k ≤ 6 , PSU 3 ( 3 ) , PSU 3 ( 3 2 ) , PSU 3 ( 3 3 ) , PSU 3 ( 5 ) , PSU 3 ( 7 ) , PSU 4 ( 2 k ) for 1 ≤ k ≤ 3 , PSU 4 ( 3 ) , PSU 5 ( 2 ) , PSU 6 ( 2 ) , P Ω 7 ( 3 ) , P Ω 9 ( 2 ) , P Ω 8 + ( 2 ) , P Ω 8 − ( 2 ) , PS p 6 ( 3 ) , PS p 8 ( 2 ) , PS p 4 ( 2 k ) for 2 ≤ k ≤ 4 , PS p 4 ( 3 ) , PS p 4 ( 3 2 ) , PS p 4 ( 5 ) , PS p 6 ( 2 ) , PS p 6 ( 4 ) .

Then, computation shows that T is in Part 3 of Table 4.□

The next lemma is about the vertex stabilizer in an arc-transitive group of automorphisms of symmetric graph of valency 11, see [10] and [13].

Lemma 2.7

Let Γ be a connected G-arc-transitive graph with valency 11 and α a vertex of Γ . Then, one of the following statements holds:

  1. If G α is soluble, then ∣ G α ∣ ∣ 1,100 and G α is one of

    Z 11 , D 22 , F 55 , Z 2 × D 22 , Z 5 × D 55 , F 110 , Z 2 × F 110 , Z 5 × F 110 , Z 110 × F 110 .

  2. If G α is insoluble, then ∣ G α ∣ ∣ 2 16 ⋅ 3 8 ⋅ 5 4 ⋅ 7 2 ⋅ 11 , and the pairs ( G α , ∣ G α ∣ ) lie in Table 5.

Table 5

Insoluble vertex stabilizer of arc-transitive graph with valency 11

G α ∣ G α ∣ G α ∣ G α ∣ G α ∣ G α ∣
PSL(2,11) 2 2 ⋅ 3 ⋅ 5 ⋅ 11 M 11 2 4 ⋅ 3 2 ⋅ 5 ⋅ 11 A 11 2 7 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ 11
S 11 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ 11 A 5 × PSL ( 2 , 11 ) 2 4 ⋅ 3 2 ⋅ 5 ⋅ 11 A 6 × M 11 2 7 ⋅ 3 4 ⋅ 5 2 ⋅ 11
M 10 × M 11 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 11 ( A 10 × A 11 ) : Z 2 2 15 ⋅ 3 8 ⋅ 5 4 ⋅ 7 2 ⋅ 11 A 10 × A 11 2 14 ⋅ 3 8 ⋅ 5 4 ⋅ 7 2 ⋅ 11
S 10 × S 11 2 16 ⋅ 3 8 ⋅ 5 4 ⋅ 7 2 ⋅ 11

A typical method for studying vertex-transitive graphs is taking normal quotients. Let Γ be a G -vertex-transitive graph, where G ≤ Aut Γ . Suppose that G has a normal subgroup N , which is intransitive on V Γ . Let V Γ N be the set of N -orbits on V Γ . The normal quotient graph Γ N of Γ induced by N is defined as the graph with vertex set V Γ N , and B is adjacent to C in Γ N if and only if there exist vertices β ∈ B and γ ∈ C such that β is adjacent to γ in Γ . In particular, if val ( Γ ) = val ( Γ N ) , then Γ is called a normal cover of Γ N .

A graph Γ is called G-locally primitive if, for each α ∈ V Γ , the stabilizer G α acts primitively on Γ ( α ) . Obviously, an arc-transitive pentavalent graph is locally primitive. The following theorem gives a basic method for studying vertex-transitive locally primitive graphs, see [14, Theorem 4.1] and [15, Lemma 2.5].

Theorem 2.8

Let Γ be a G-vertex-transitive locally primitive graph, where G ≤ Aut Γ , and let N ◃ G have at least three orbits on V Γ . Then, the following statements hold:

  1. N is semi-regular on V Γ , G / N ≤ Aut Γ N , and Γ is a normal cover of Γ N ;

  2. G α ≅ ( G / N ) γ , where α ∈ V Γ and γ ∈ V Γ N ;

  3. Γ is ( G , s ) -transitive if and only if Γ N is ( G / N , s ) -transitive, where 1 ≤ s ≤ 5 or s = 7 .

For the case where N has at most two orbits on V Γ , the next fact is a consequence of the connectivity of the graph, which is well known.

Lemma 2.9

Let Γ be a connected G-arc-transitive graph of odd prime valency d . Let 1 ≠ N be a normal subgroup of G. Suppose that N have at most two orbits on V Γ and N α ≠ 1 , where α is a vertex of Γ . Then, N α is transitive on the neighbors Γ ( α ) of α , particularly, d ∣ ∣ N α ∣ .

By Li and Feng [16, Theorem 3.6], we have the following lemma.

Lemma 2.10

Let n be a square-free integer and Γ a 7-valent one-regular graph of order n. Then, n = 2 ⋅ 7 t ⋅ p 1 p 2 ⋯ p s ≥ 13 , where t ≤ 1 , s ≥ 1 , and p i ’s are distinct primes such that 7 ∣ ( p i − 1 ) . Furthermore, Γ is isomorphic to one of CD n l and there are exactly 6 s − 1 such non-isomorphic graphs of order n.

For reduction, we need some information of 7-valent symmetric graphs of order 2 p , stated in the following lemma, see [2, Table 1].

Lemma 2.11

Let p be a prime and let Γ be a 7-valent symmetric graph of order 2 p . Then, Γ is isomorphic to one of the following graphs:

  1. The complete bipartite graph K 7 , 7 for p = 7 with Aut Γ ≅ S 7 ≀ S 2 .

  2. The graph G ( 2 p , 7 ) for p > 7 with Aut G ( 2 p , 7 ) ≅ D 2 p : Z 7 .

Remark of Lemma 2.11. We define the graph G ( 2 p , 7 ) in the following. Let A and A ′ be two disjoint copies of Z p . For each element i of Z p , we shall denote the corresponding elements of A and A ′ by i and i ′ , respectively. Let r be a positive integer dividing p − 1 , where p is prime, and let H ( p , r ) denote the unique subgroup of Z p ∗ of order r . We define the graph G ( 2 p , r ) to have vertex-set A ∪ A ′ and edge-set { x y ′ : x , y ∈ Z p , and y − x ∈ H ( p , r ) } .

We need some classification results on symmetric graphs of valency 11. The following two lemmas are obtained from [2], [17], and [18].

Lemma 2.12

Let Γ be a connected symmetric graph of order 2 r and valency 11, where r is an odd prime. Suppose that Aut Γ is insolvable, then Γ is the complete bipartite graph K 11 , 11 .

Lemma 2.13

Let Γ be a connected symmetric graph of order 2 m and valency 11, where m is an odd square-free integer, then one of the following statements holds:

  1. Γ is a normal Cayley graph on D 2 m and Aut Γ = D 2 m : Z 11 ;

  2. Aut Γ = J 1 , Aut Γ α = PSL ( 2 , 11 ) , and m = 7 ⋅ 19 , moreover, Γ is not bipartite;

  3. Aut Γ = PSL ( 2 , r ) or PGL ( 2 , r ) where r ≡ ± 1 ( mod 11 ) is a prime.

3 Examples

In this section, we give some examples of 7-valent symmetric graphs of order 2pq with q > p ≥ 3 distinct primes.

For a given small permutation group X , one may determine all graphs that admit X as an arc-transitive automorphism group by using Magma program [19]. It is then easy to have the following result.

Example 3.1

There are two connected 7-valent symmetric graphs of order 78, which admit PSL ( 2 , 13 ) or PGL ( 2 , 13 ) as an arc-transitive automorphism group. These two graphs are denoted by C 78 1 and C 78 2 , which satisfy the conditions in Rows 1 and 2 of Table 1.

Example 3.2

There is a unique connected 7-valent symmetric graph of order 310, which admits PSL ( 5 , 2 ) . Z 2 as an arc-transitive automorphism group. This graph is denoted by C 310 , which satisfies the conditions in Row 3 of Table 1.

Example 3.3

There is a unique connected 7-valent symmetric graph of order 30, which admits S 8 as an arc-transitive automorphism group. This graph is denoted by C 30 , which satisfies the conditions in Row 4 of Table 1.

4 The proof of Theorem 1.2

Now, we prove the main result of this article. Let Γ be a 7-valent symmetric graph of order 2pq. Set A = Aut Γ . By Lemma 2.4, ∣ A α ∣ ∣ 2 24 ⋅ 3 4 ⋅ 5 2 ⋅ 7 , and hence ∣ A ∣ ∣ 2 25 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ p ⋅ q . Let R be the soluble radical of A and let F be the Fitting subgroup of A (recall that the Fitting subgroup F of A is defined to be the product of all normal nilpotent subgroups of A ). We divide our discussion into the following three cases.

Case 1. R = 1

Let N be a minimal normal subgroup of A and let C = C A ( N ) . Since R = 1 , we have that N = T d , where T is a nonabelian simple group and d ≥ 1 . Furthermore, since ∣ A ∣ ∣ 2 25 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ p ⋅ q , we have ∣ N ∣ ∣ 2 25 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ p ⋅ q .

Assume that N has t orbits on V Γ . If t ≥ 3 , then by Theorem 2.8, N α = 1 and so ∣ N ∣ = ∣ T ∣ d ∣ 2 p q , which is a contradiction as T is a nonabelian simple group. Hence, N α ≠ 1 , N has at most two orbits on V Γ and p q divides ∣ N : N α ∣ . Since Γ is connected, N ◃ A , and N α ≠ 1 , we have 1 ≠ N α Γ ( α ) ◃ A α Γ ( α ) , it follows that 7 divides ∣ N α ∣ , we thus have that 7 p q divides ∣ T ∣ .

We first show that d = 1 . If not, d ≥ 2 , then 7 2 ∣ ∣ T ∣ 2 = ∣ N ∣ as ∣ N ∣ = ∣ T ∣ d ∣ 2 25 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ p ⋅ q . It follows that p = 7 or q = 7 . If p = 7 , then q > 7 and q 2 ∣ ∣ T ∣ 2 , a contradiction. If q = 7 , then p = 3 or 5. It can conclude that ∣ T ∣ ∣ 2 12 ⋅ 3 2 ⋅ 5 ⋅ 7 . Note that 21 ∣ ∣ T ∣ or 35 ∣ ∣ T ∣ . By checking the nonabelian simple group of order less than 2 12 ⋅ 3 2 ⋅ 5 ⋅ 7 (e.g., [12]), we have that T ≅ A 7 , A 8 , or PSL ( 3 , 4 ) , and so d = 2 , N = A 7 2 , A 8 2 , or PSL ( 3 , 4 ) 2 . On the other hand, C ◃ A , C ∩ N = 1 and ⟨ C , N ⟩ = C × N . Because ∣ C × N ∣ divides 2 25 ⋅ 3 4 ⋅ 5 2 ⋅ 7 2 ⋅ p ⋅ q and ∣ N ∣ = ∣ T ∣ 2 = 2 6 ⋅ 3 4 ⋅ 5 2 ⋅ 7 2 or 2 9 ⋅ 3 4 ⋅ 5 2 ⋅ 7 2 , C is a { 2 , p } -group, and hence soluble, where p = 3 or p = 5 . So C = 1 as R = 1 . This implies A = A / C ≤ Aut ( N ) ≅ Aut ( T ) ≀ S 2 . By Magma [19], no such graph exists. Thus, d = 1 and N = T ⊴ A is a nonabelian simple group.

We next show that C = 1 . If not, then C is insoluble as R = 1 and C ⊴ A . The same argument as for the case N leads to 7 ∣ ∣ C α ∣ . Since ⟨ C , N ⟩ = C × N and C , N ⊴ A , we have N α × C α ≤ A α . On the other hand, 7 ∣ ∣ N α ∣ , it concludes that 7 2 ∣ ∣ A α ∣ , a contradiction with Lemma 2.4. Hence, A is almost simple and A ≤ Aut ( T ) . Thus, we have soc ( A ) = T as a nonabelian simple group and satisfies the following condition.

Condition ( ∗ ) : ∣ T ∣ lies in Table 3 such that 7 p q ∣ ∣ T ∣ and ∣ T ∣ ∣ 2 25 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ p ⋅ q .

Assume first that T = M 22 , M 24 , J 1 , J 2 , HS , PSU ( 3 , 8 ) , PSp ( 6 , 2 ) , PSp ( 8 , 2 ) , PSp ( 4 , 8 ) , PSL ( 3 , 4 ) , PSL ( 2 , 2 9 ) , PSL ( 2 , 27 ) , PSL ( 2 , 125 ) , PSL ( 2 , 49 ) , PSU ( 3 , 5 ) , PSL ( 3 , 16 ) , A 9 , or A 10 . Note that ∣ T : T α ∣ = p q or 2pq. By Atlas [20], T has no subgroup of index p q or 2pq, a contradiction.

Assume that T = M 23 , A 7 , A 11 , A 12 , Sz ( 8 ) , PSL ( 4 , 2 ) , PSL ( 4 , 4 ) , PSL ( 3 , 8 ) , or PSL ( 2 , 2 6 ) . Note that T ≤ A ≤ Aut ( T ) . We can exclude all these cases by using Magma [19].

Assume that T = PSL ( 5 , 2 ) . Then, ( p , q ) = ( 3 , 31 ) or ( 5 , 31 ) . For the former case, T has no subgroup of index 93 or 186, a contradiction. For the latter case, by Example 3.2, Γ is isomorphic to C 310 . Assume that T = A 8 . Then, ( p , q ) = ( 3 , 5 ) . By Example 3.3, Γ is isomorphic to C 30 .

Assume that T = PSL ( 2 , q ) . Then, T ≤ A ≤ Aut ( T ) = PGL ( 2 , q ) and ∣ A : T ∣ ≤ 2 . If A α is insoluble, then T α is also insoluble as ∣ A α : T α ∣ ≤ 2 . By Lemma 2.2, T α = A 5 , which is impossible as 7 ∣ ∣ T α ∣ . Thus, A α is soluble, and by Lemma 2.4, A α divides 252, and so ∣ T α ∣ ∣ 252 . It implies that the order of T divides 504 ⋅ p ⋅ q . Note that ∣ PSL ( 2 , q ) ∣ = q ( q − 1 ) ( q + 1 ) 2 and ( q + 1 2 , q − 1 2 ) = 1 . If p ∣ q − 1 2 , then q + 1 divides 504. It follows that q = 5 , 7 , 11 , 13 , 17 , 23 , 41 , 71 , 83 , 167 , 251 , or 503. However, PSL ( 2 , q ) does not satisfy the Condition ( * ) for q = 5 , 7 , 11 , 17 , or 23. Thus, q = 13 , 41 , 71 , 83 , 167 , 251 , or 503 for this case. If p ∣ q + 1 2 , then q − 1 divides 504. It follows that q = 5 , 7 , 13 , 19 , 29 , 37 , 43 , 73 , or 127. However, PSL ( 2 , q ) does not satisfy the Condition ( * ) for q = 5 , 7 , 19 , 37 , or 73. Thus, q = 13 , 29 , 43 , or 127 for this case. Therefore, for T = PSL ( 2 , q ) , T is one of the following groups:

T Order T Order
PSL ( 2 , 13 ) 2 2 ⋅ 3 ⋅ 7 ⋅ 13 PSL ( 2 , 29 ) 2 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 29
PSL ( 2 , 41 ) 2 3 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 41 PSL ( 2 , 43 ) 2 2 ⋅ 3 ⋅ 7 ⋅ 11 ⋅ 43
PSL ( 2 , 71 ) 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 71 PSL ( 2 , 83 ) 2 2 ⋅ 3 ⋅ 7 ⋅ 41 ⋅ 83
PSL ( 2 , 127 ) 2 7 ⋅ 3 2 ⋅ 7 ⋅ 127 PSL ( 2 , 167 ) 2 3 ⋅ 3 ⋅ 7 ⋅ 83 ⋅ 167
PSL ( 2 , 251 ) 2 2 ⋅ 3 2 ⋅ 5 3 ⋅ 7 ⋅ 251 PSL ( 2 , 503 ) 2 3 ⋅ 3 2 ⋅ 7 ⋅ 251 ⋅ 503

Assume that q = 29 , 41 , 71 , 127 , or 251. Note that ∣ T : T α ∣ = p q or 2pq. By Lemma 2.2, T has no subgroup of index p q or 2pq, a contradiction. Assume that q = 43 , 83 , or 167. Note that A = PGL ( 2 , q ) or PSL ( 2 , q ) . We can exclude all the cases by Magma [19]. Assume that q = 503 . Then, ∣ T α ∣ = 504 or 252. It implies that T α is soluble and so as A α . By Lemma 2.4, ∣ A α ∣ ∣ 252 , and therefore, A = PSL ( 2 , 503 ) , ∣ A α ∣ = 252 . Again by Lemma 2.4, A α ≅ F 42 × Z 6 , which is impossible by Lemma 2.2. Assume, finally, that q = 13 . Then, T = PSL ( 2 , 13 ) and A = PSL ( 2 , 13 ) or PGL ( 2 , 13 ) . By example 3.1, Γ is isomorphic to C 78 1 or C 78 2 . This completes the proof of this case.

Case 2. R ≠ 1 and A is soluble

Then, R = A , and by Lemma 2.1, F ≠ 1 and C A ( F ) ≤ F . As ∣ V Γ ∣ = 2 p q , A has no nontrivial normal Sylow s -subgroup, where s ≠ 2 , p , or q . So F = O 2 ( A ) × O p ( A ) × O q ( A ) , where O 2 ( A ) , O p ( A ) , and O q ( A ) denote the largest normal 2-, p -, and q -subgroups of A , respectively.

For each r ∈ { 2 , p , q } , since q > p ≥ 3 , O r ( A ) has at least three orbits on V Γ , by Proposition 2.8, O r ( A ) is semi-regular on V Γ . Therefore, ∣ O 2 ( A ) ∣ ≤ 2 , ∣ O p ( A ) ∣ ≤ p , ∣ O q ( A ) ∣ ≤ q , F ≤ Z 2 p q is abelian, and C R ( F ) = F .

If ∣ F ∣ = 2 , by Proposition 2.8, the normal quotient graph Γ F is a 7-valent A / F -arc-transitive graph of odd order p q , not possible. Thus, there exists a prime r ∈ { p , q } such that r ∣ ∣ F ∣ , and so O r ( A ) = r . By Theorem 2.8, Γ O r ( A ) is a 7-valent A / O r ( A ) -arc transitive graph of order 2 s with s ∈ { p , q } and A / O r ( A ) is soluble. Then, by Lemma 2.11, Γ O r ( A ) is isomorphic to K 7 , 7 or G ( 2 p , 7 ) . For the former case, by [21, Theorem 1.1], p = 7 and Γ O r ( A ) ≅ CD 14 q k as described in Theorem 1.2 (1). For the latter case, by Lemma 2.11, Γ O r ( A ) ≅ G ( 2 p , 7 ) and Aut Γ O r ( A ) ≅ D 2 s : Z 7 is arc-regular on A Γ . Hence, A / O r ( A ) ≅ D 2 s : Z 7 , it implies that Γ is an 7-valent arc-regular graph of order 2pq. By Lemma 2.10, Γ ≅ CD 2 p q k as in Theorem 1.2 (1).

Case 3. R ≠ 1 and A is insoluble

Let N be a minimal soluble normal subgroup of A . Then, N ≅ Z r d has at least three orbits on V Γ , where r is a prime. It follows from Theorem 2.8 that N is semi-regular on V Γ , and so d = 1 , r ∈ { p , q } . Furthermore, Γ N is A / N -arc-transitive graph of order 2 p q r = 2 t and A / N is insoluble, where t ∈ { p , q } . Since Γ N is A / N -arc-transitive and A / N is insoluble, by Lemma 2.11, Γ N is isomorphic to K 7 , 7 . Thus, Γ is a normal Z t -cover of K 7 , 7 , where t ≠ 7 . By [21, Theorem 1.1], no such graph Γ exists.

Thus, we complete the proof of Theorem 1.2.

5 The proof of Theorems 1.3 and 1.4

In this section, we prove Theorems 1.3 and 1.4. Let Γ be a connected symmetric graph of order 4 n and valency 11, where n = p ⋅ q with p , q ≥ 3 two distinct primes, and let α be a vertex of Γ . Set A = Aut Γ and let R be the largest solvable normal subgroup of A .

Lemma 5.1

A is insolvable.

Proof

Suppose for a contradiction that A is solvable. Let H be the Fitting subgroup of A . Then, H is nilpotent and H is the product of all its Sylow r -subgroups, where r is a prime dividing ∣ H ∣ . Clearly, H r is characteristic in H , and hence, normal in A . If H r has at most two orbits on V Γ , then 2 n = ∣ V Γ ∣ / 2 divides ∣ H r ∣ , a contradiction. Therefore, H r has at least three orbits on V Γ . Considering the quotient graph Γ H r , by Lemma 2.8, we have H r is semi-regular on V Γ , and hence ∣ H r ∣ ∣ 4 n , and Γ H r is a connected A / H r -arc-transitive graph of valency 11. This implies ∣ H 2 ∣ = 1 or 2 as there is no symmetric graph of odd order and odd valency, and ∣ H r ∣ is a prime if r is odd. Then, H is cyclic. Let C = C A ( H ) . Then, C ≤ H by Lemma 2.1, and hence, C = H . Thus, A / H = A / C ≤ Aut ( H ) is abelian. Since A α ≅ A α / ( A α ∩ H ) ≅ H A α / H ≤ A / H is abelian, A α is abelian, and hence A α ≅ Z 11 by Lemma 2.7. Thus, Γ is an arc-regular graph (i.e., Aut Γ is regular on the arc set of Γ ). However, there is no arc-regular graph of order four times an odd square-free integer, see [22], a contradiction. This proves the lemma.□

Lemma 5.2

Assume that A is insolvable and R = 1 . Then, T ⊴ A ≤ Aut ( T ) for some nonabelian simple group T, and T has at most two orbits on V Γ and 11 ∣ ∣ T α ∣ .

Proof

Let N ≠ 1 be a minimal normal subgroup of A . Then, N = T m for some nonabelian simple group T , where m ≥ 1 be a positive integer. Since T is nonabelian simple group, 4 ∣ ∣ T ∣ , and so 4 m ∣ ∣ N ∣ . If N has at least three orbits on V Γ , then N α = 1 by Lemma 2.8. Considering the quotient graph Γ N , and 4 ∣ ∣ V Γ ∣ = 4 n , we obtain the quotient graph Γ N is of odd order and valency 11, which is impossible. Therefore, N has at most two orbits on V Γ .

If N α = 1 , then the order of N divides ∣ V Γ ∣ = 4 n . Since 4 ∣ ∣ T ∣ , we have N = T . Note that ∣ T ∣ has at least three prime divisors. Thus, n = p q and ∣ T ∣ = 4 p q . By [23], we have N ≅ A 5 . Note that now N is regular on V Γ . Thus, Γ = Cay ( N , S ) is a normal Cayley graph for some subset S ⊆ N ⧹ { 1 } . Then, by Lemma 2.3, A α ≅ A 1 = Aut ( N , S ) ≤ Aut ( N ) ≅ S 5 , and so 11 ∤ ∣ A α ∣ , contradicting to Lemma 2.7.

Therefore, N α ≠ 1 . Then, by Lemma 2.9, 11 ∣ ∣ N α ∣ . If A has another minimal normal subgroup, namely M , then 11 ∣ ∣ M α ∣ by an argument similar to N . It follows 1 1 2 ∣ ∣ M α × N α ∣ ∣ ∣ A α ∣ , a contradiction. Therefore, N is the unique minimal normal subgroup of A .

It remains to show N = T . Since N has at most two orbits on V Γ , n ∣ ∣ N ∣ . This implies 4 ⋅ 11 ⋅ n ∣ ∣ T ∣ . Thus, ( 4 ⋅ n ) m ⋅ 1 1 m ∣ ∣ N ∣ ∣ ∣ A ∣ = 4 n ∣ A α ∣ . Then, ( 4 ⋅ n ) m − 1 ⋅ 1 1 m ∣ ∣ A α ∣ . Since 1 1 2 ∤ ∣ A α ∣ by Lemma 2.7, we have m = 1 . Thus, N = T , as required.□

We further determine graphs in the case where A is insolvable and R = 1 .

Lemma 5.3

Assume that A is insolvable and R = 1 .

  1. If n is a prime, then Γ = K 12 .

  2. If n = p q , where p > q ≥ 3 are two distinct primes, then ( Γ , A , A α ) is listed in the first three rows of Table 2.

Proof

By Lemma 5.3, T ⊴ A ≤ Aut ( T ) for a nonabelian simple group, T has at most two orbits on V Γ , and 11 ∣ ∣ T α ∣ . By Lemma 2.7, ∣ A ∣ = ∣ V Γ ∣ ∣ A α ∣ ∣ 2 18 ⋅ 3 8 ⋅ 5 4 ⋅ 7 2 ⋅ 11 ⋅ n . Thus, 44 ∣ ∣ T ∣ and ∣ T ∣ ∣ 2 18 ⋅ 3 8 ⋅ 5 4 ⋅ 7 2 ⋅ 11 ⋅ n . Therefore, such simple groups T are determined by Lemma 2.6 and are listed in Table 4.

We first deal with the case where T = PSL ( 2 , p ) . Assume that T = PSL ( 2 , p ) . Then, A = PSL ( 2 , p ) or PGL ( 2 , p ) . By the information of maximal subgroups of PSL ( 2 , p ) and PGL ( 2 , p ) in Lemma 2.2, we know A α is solvable. Then, by Lemma 2.7, there are only three possibilities for A α , that are Z 11 , D 22 , and D 22 × Z 2 . In particular, A α is a { 2 , 11 } -group. Since ∣ A ∣ = 4 n ∣ A α ∣ , where n has at most two prime divisors, we obtain ∣ A ∣ that has at most four prime divisors, so does ∣ T ∣ . By [23], T = PSL ( 2 , 23 ) . Noting ∣ P S L ( 2 , 23 ) ∣ = 2 3 ⋅ 3 ⋅ 11 ⋅ 23 and p > q , we have p = 23 and q = 3 . Then, by Lemma 2.7, pair ( A , A α ) = ( PGL ( 2 , 23 ) , D 44 ) or ( PSL ( 2 , 23 ) , D 22 ) . Computation with Magma [19] shows that, up to graph isomorphism, there is only one such graph Γ for pair ( PGL ( 2 , 23 ) , D 44 ) , say G 276 1 , with automorphism group PGL ( 2 , 23 ) ; and there are three graphs for pair ( PSL ( 2 , 23 ) , D 22 ) , say G 276 i for 2 ≤ i ≤ 4 , with automorphism group PSL ( 2 , 23 ) . Then, ( Γ , A , A α ) is as the second and third rows of Table 2.

Now, we prove parts (1) and (2) of the lemma.

(1). Suppose that n is a prime.

Clearly, n ≠ 2 . If n = 3 , then ∣ V Γ ∣ = 12 , which implies that Γ = K 12 . Actually, Γ = K 12 arises when T = M 11 , M 12 , A 12 , P S L ( 2 , 11 ) because each of them has a 2-transitive permutation representation of degree 12 (see [24]).

Therefore, we assume that n ≥ 5 . Since T has at most two orbits on V Γ , ∣ T : T α ∣ = 2 n or 4 n . By Atlas [20] or direct computation in Magma [19], it is easy to check whether a simple group T in Table 4 has a subgroup of index 2 n or 4 n and of order divisible by 11. For example, let T = M 11 , then Atlas [20] tells us that a maximal subgroup of M 11 has index 11 , 12 , 55 , 60 , and 165 and the maximal subgroup of index 11 is M 10 = A 6 . 2 . Therefore, the only possibility for T α is A 6 . However, this contradicts 11 ∣ ∣ T α ∣ . Therefore, we can rule out the case T = M 11 . Other simple groups can be ruled out similarly.

(2). Suppose that n = p q , where p > q ≥ 3 are two distinct primes.

We may assume that T ≠ PSL ( 2 , p ) ( p > 11 ) . Assume T ≅ M 11 with order 2 4 ⋅ 3 2 ⋅ 5 ⋅ 11 . Then, A = M 11 as Out ( M 11 ) = 1 . Then, p q = 5 ⋅ 3 and A α is of order 2 2 ⋅ 3 ⋅ 11 , contradicting Lemma 2.7. By an argument similar to M 11 , we can rule out M 23 , M 24 , J 1 , C o 2 , and C o 3 , those simple groups with outer automorphism group 1.

Assume N ≅ M 12 with order 2 6 ⋅ 3 3 ⋅ 5 ⋅ 11 . Then, A ≅ M 12 or M 12 . Z 2 as Out ( M 12 ) = 2 , refer to [20]. Then, p q = 5 ⋅ 3 and A α is of order 2 4 ⋅ 3 2 ⋅ 11 or 2 5 ⋅ 3 2 ⋅ 11 . By Lemma 2.7, this is impossible. Similarly, we can rule out other simple groups with nontrivial out automorphism groups in Table 4, except two groups PSL ( 2 , 11 ) and PSL ( 2 , 1 1 2 ) .

Assume T = PSL ( 2 , 11 ) with order 2 2 ⋅ 3 ⋅ 5 ⋅ 11 . Then, A = PSL ( 2 , 11 ) or PGL ( 2 , 11 ) as Out ( PSL ( 2 , 11 ) ) ≅ Z 2 . In this case, p q = 5 ⋅ 3 , and hence, ∣ V Γ ∣ = 60 . Computation in Magma shows that there is a unique such graph Γ up to graph isomorphism, which is G 60 , its automorphism group and the vertex of stabilizer are PGL ( 2 , 11 ) and D 22 , respectively. This is the first row of Table 2.

Assume T = PSL ( 2 , 1 1 2 ) . Then, A = PSL ( 2 , 1 1 2 ) . o , where o ≤ Out ( PSL ( 2 , 1 1 2 ) ) ≅ Z 2 2 . Note that ∣ P S L ( 2 , 1 1 2 ) ∣ = 2 3 ⋅ 3 ⋅ 5 ⋅ 1 1 2 ⋅ 61 . By Lemma 2.7, 11 is the largest prime divisor of A α , thus p q = 61 ⋅ 11 and so ∣ V Γ ∣ = 4 ⋅ 11 ⋅ 61 . If o = 1 , then A = PSL ( 2 , 1 1 2 ) with ∣ A α ∣ = 2 ⋅ 3 ⋅ 5 ⋅ 11 , contradicting Lemma 2.7. If o = Out ( PSL ( 2 , 1 1 2 ) ) ≅ Z 2 2 , then ∣ A α ∣ = 2 3 ⋅ 3 ⋅ 5 ⋅ 11 , also contradicting Lemma 2.7. Therefore, o ≅ Z 2 . Then, ∣ A α ∣ = 2 2 ⋅ 3 ⋅ 5 ⋅ 11 , and so A α = PSL ( 2 , 11 ) by Lemma 2.7. However, no such graph exists by computation with Magma [19].□

At last, we complete the proof of Theorems 1.3 and 1.4 by dealing with the case where R ≠ 1 .

Lemma 5.4

Assume that A is insolvable and R ≠ 1 . Then, p = 19 , q = 7 , and ( Γ , A , A α ) = ( G 532 , Z 2 × J 1 , PSL ( 2 , 11 ) ) , as the fourth row of Table 2.

Proof

Since R ≠ 1 , A has a minimal normal subgroup N ≅ Z r m ≠ 1 contained in R , where r is a prime. If N has at most two orbits on V Γ , then 2 n = ∣ V Γ ∣ / 2 ∣ ∣ N ∣ , a contradiction. Therefore, N has at least three orbits on V Γ . Considering the quotient graph Γ N , by Lemma 2.8, we have N as semi-regular on V Γ , and hence, ∣ N ∣ ∣ 4 n and Γ N is a connected A / N -arc-transitive graph of valency 11. Put v a vertex of V Γ N .

Case 1. Suppose that n is a prime.

Then, we have r = 2 . Then, N = Z 2 because if N = Z 2 2 , then Γ N is a symmetric graph of odd order and valency 11, which is impossible. Note that A / R is insolvable as A is insolvable. By Lemma 2.12, Γ N = K 11 , 11 , and so Γ is a normal Z 2 -cover of K 11 , 11 . However, there is no such graph Γ by [21, Theorem 1.1]. This proves Theorem 1.3.

Case 2. Suppose that n = p q , where p > q ≥ 3 are two distinct primes.

Then, r ∈ { p , q , 2 } as ∣ N ∣ ∣ 4 p q . If r = q , then ∣ V Γ N ∣ = 4 p . By Theorem 1.3 Γ N = K 12 , and so p = 3 , contradicting p > q ≥ 3 . Therefore, r ≠ q .

Assume that r = p . Then, ∣ V Γ N ∣ = 4 q . By Theorem 1.3 Γ N = K 12 , and so q = 3 . Note that a subgroup H ≤ Aut Γ N = S 12 is arc-transitive if and only if H is 2-transitive on 12 points. By the classification of 2-transitive permutation groups, see, e.g., [24], possibilities for ( A / N , ( A / N ) v ) are as follows:

( PSL ( 2 , 11 ) , Z 11 : Z 5 ) , ( PGL ( 2 , 11 ) , Z 11 : Z 10 ) , ( M 11 , PSL ( 2 , 11 ) ) , ( A 12 , A 11 ) , ( S 12 , S 11 ) .

If ( A / N , ( A / N ) v ) ≠ ( PSL ( 2 , 11 ) , Z 11 : Z 5 ) , then A / N acts 2-arc-transitively on Γ N = K 12 . Note that 2-arc-transitive cyclic cover of complete graph was determined in [25, Theorem 1.1], and from their result we can obtain a contradiction.

Therefore, ( A / N , ( A / N ) v ) = ( PSL ( 2 , 11 ) , Z 11 : Z 5 ) . Then, A = Z p . P S L ( 2 , 11 ) . Since the Schur multiplier of PSL ( 2 , 11 ) is isomorphic to Z 2 , see Atlas [20], we have A = N A ′ = N × A ′ = Z p × PSL ( 2 , 11 ) . Let K = PSL ( 2 , 11 ) ⊴ A . Note that ∣ P S L ( 2 , 11 ) ∣ = 2 2 ⋅ 3 ⋅ 5 ⋅ 11 . Therefore, K α ≠ 1 . Then, Lemma 2.7 implies that K has at most two orbits on V Γ . By Lemma 2.9, 11 ∣ ∣ K α ∣ . Then p = 5 . Since K has at most two orbits on V Γ , ∣ K α ∣ = ∣ K ∣ / ( 2 p q ) = 44 or ∣ K ∣ / ( 4 p q ) = 22 . By Atlas [20], PSL ( 2 , 11 ) has no subgroup of order 44 but has subgroups isomorphic to D 22 of order 22. Therefore, K α = D 22 and K is transitive on V Γ , and hence Γ is K -arc-transitive. By computation in Magma [19], Γ = G 60 with automorphism group PGL ( 2 , 11 ) , contradicting the assumption that R ≠ 1 .

Assume last that r = 2 . Then, N = Z 2 and Γ N satisfies Lemma 2.13. Since A / N is insolvable, the case (1) of Lemma 2.13 is impossible.

Suppose that case (2) of Lemma 2.13 happens, that is, p = 19 , q = 7 , Aut Γ N = J 1 , and ( Aut Γ N ) v = PSL ( 2 , 11 ) . Now, A / N ≤ Aut Γ N = J 1 . Note that ∣ J 1 ∣ = 2 3 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 19 . Since A / N acts arc-transitively on Γ N , ∣ A / N ∣ is divisible by 11 ⋅ ∣ V Γ N ∣ = 11 ⋅ 2 ⋅ 7 ⋅ 19 . By the information of maximal subgroups of J 1 in Atlas [20], we have A / N = J 1 . Then, A = Z 2 . J 1 . Since the Schur multiplier of J 1 is 1, also refer to Atlas [20], we have A = Z 2 × J 1 . Let K = J 1 ⊴ A . Then, K α ≠ 1 , and Lemma 2.8 implies that K has at most two orbits on V Γ . If K has two orbits on V Γ , then ∣ K α ∣ = ∣ K ∣ / 2 p q = 660 ; however, J 1 has no subgroup of order 660 by Atlas [20]. Thus, K is transitive on V Γ , and hence Γ is arc-transitive by Lemma 2.9. Computation in Magma [19] shows that there is a unique such graph Γ , which is G 532 , with automorphism group Z 2 × J 1 .

Suppose that case (3) of Lemma 2.13 happens, then Aut Γ N = PSL ( 2 , r ) or PGL ( 2 , r ) , where r is a prime such that r ≡ ± 1 ( mod 11 ) . By Lemma 2.2, we have PSL ( 2 , r ) ≤ A / N as A / N is insolvable. In addition, by Lemma 2.2, we have ( A / N ) v = Z 11 , D 22 or Z 2 × D 22 . Then, PSL ( 2 , r ) is a simple group with at most four prime divisors and 11 ∣ ∣ P S L ( 2 , r ) ∣ , and hence r = 23 by [23]. Then, we obtain p = 23 and q = 3 . If A / N = PSL ( 2 , 23 ) , then ∣ ( A / N ) v ∣ = ∣ P S L ( 2 , 23 ) ∣ / ( 2 ⋅ 3 ⋅ 23 ) = 44 ; however, PSL ( 2 , 23 ) has no subgroup of order 44 by Lemma 2.2, a contradiction. If A / N = PGL ( 2 , 23 ) , then ∣ ( A / N ) v ∣ = ∣ A α ∣ = 88 ; however, PGL ( 2 , 23 ) has no subgroup of order 88, see Lemma 2.2, which is a contradiction.□

Acknowledgement

The authors are grateful to the constructive comments of referees.

  1. Funding information: This work was partially supported by the National Natural Science Foundation of China (12061089, 11861076, 11701503, and 11761079), and the Natural Science Foundation of Yunnan Province (202201AT070022, 2018FB003, and 2019FB139).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state that there is no conflict of interest.

  4. Data availability statement: Not applicable.

Appendix A

Magma codes used in Example 3.1

TU:=[];
j:=0;
G:=PSL(2,13);
H:=Subgroups(G:OrderEqual:=14);
for t in [1..#H] do
HH:=H[t]‘subgroup;
A:=CosetAction(G,HH);
O:=Orbits(A(HH));
for i in [1..#O] do
OO:=SetToSequence(O[i]); GA:=OrbitalGraph(A(G),1,OO[1]);
if (IsConnected(GA) eq true) and (Valence(GA) eq 7) and
(not existst:t in TU∣IsIsomorphic(GA,t) eq true) then
Append( TU,GA);
j:=j+1;
end if;
end for;
end for;
j;

References

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Received: 2021-10-20
Revised: 2022-07-21
Accepted: 2022-11-07
Published Online: 2022-12-31

© 2022 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
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  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
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  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
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  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
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  59. On the general position number of two classes of graphs
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  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
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  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
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  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
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  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
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  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
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  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
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  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
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  107. Study on r-truncated degenerate Stirling numbers of the second kind
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  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
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  115. Finite groups whose maximal subgroups of even order are MSN-groups
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  122. Admissible congruences on type B semigroups
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  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
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  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
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  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0528
Keywords for this article
symmetric graph; normal quotient; automorphism group
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Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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