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Symmetric graphs of valency seven and their basic normal quotient graphs

  • Jiangmin Pan EMAIL logo , Junjie Huang and Chao Wang
Published/Copyright: August 6, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

We characterize seven valent symmetric graphs of order 2 p q n with p < q odd primes, extending a few previous results. Moreover, a consequence partially generalizes the result of Conder, Li and Potočnik [On the orders of arc-transitive graphs, J. Algebra 421 (2015), 167–186].

Keywords: symmetric graph; simple group; normal quotient graph
MSC 2010: 20B15; 20B30; 05C25

1 Introduction

In this paper, graphs considered are undirected and have no loops and multiple edges. Let Γ be a graph. We denote by V Γ and Aut Γ the vertex set and the full automorphism group of Γ , respectively. The size ∣ V Γ ∣ is called the order of Γ . For a nonnegative positive integer s , an s -arc of Γ is a sequence v 0 , v 1 , … , v s of s + 1 vertices of Γ such that v i − 1 , v i are adjacent for 1 ≤ i ≤ s and v i − 1 ≠ v i + 1 for 1 ≤ i ≤ s − 1 . If G ≤ Aut Γ is transitive on the set of s -arcs of Γ , then Γ is called ( G , s )-arc-transitive; if Γ is ( G , s ) -arc-transitive but not ( G , s + 1 ) -arc-transitive, then Γ is called ( G , s )-transitive. In particular, a 0-arc-transitive graph is called vertex-transitive, and a 1-arc-transitive graph is called arc-transitive or symmetric.

Let Γ be a G -arc-transitive graph. For a vertex intransitive normal subgroup N of G , the normal quotient graph of Γ with respect to N , denoted by Γ / N , is defined with all the N -orbits in V Γ as vertex set, and two N -orbits B , C are adjacent if and only if some vertex in B is adjacent in Γ to some vertex in C . If Γ and Γ / N have the same valency, the original graph Γ is called a normal cover of the normal quotient graph Γ / N . In particular, Γ is called basic if Aut Γ has no nontrivial normal subgroup N such that Γ and Γ / N have the same valency. By definition, there is a natural “two-step strategy” for studying symmetric graphs:

  1. Determine the normal quotient graphs of the original graphs.

  2. Reconstruct the original graphs from the normal quotient graphs by using covering techniques.

In the literature, there are a lot of contributions classifying symmetric graphs of order kpn , where k is a small positive integer and p is a prime. For example, Chao [1], Cheng and Oxley [2] and Wang and Xu [3] classified symmetric graphs of order p , order 2 p and order 3 p , respectively. For valency 3, 4 and 5 case, see [4,5, 6,7,8, 9,10,11] and references therein. Recently, a classification of 7-valent symmetric graphs of order 2 p q with p , q distinct primes has been given in [12], and a characterization of 7-valent symmetric graphs of order 4 p n was obtained in [13].

This work aims to investigate 7-valent symmetric graphs of order 2 p q n with p < q odd primes and n ≥ 2 and in particular determines all basic normal quotient graphs of such graphs, extending the results in [10,12,13]. The proof depends on the classification of finite simple groups.

Our main result is as follows. For convenience, graphs appearing in Table 1 are introduced in Section 2.

Table 1

Basic normal quotient graphs of symmetric graphs with order 2 p q n

Σ ( p , q ) Aut Σ s Σ ( p , q ) Aut Σ s
K 7 , 7 p = 7 S 7 ≀ Z 2 3 C 30 ( 3 , 5 ) S 8 2
HS ( 50 ) ( 3 , 5 ) PSU ( 3 , 5 ) . Z 2 2 C 78 1 ( 3 , 13 ) PSL ( 2 , 13 ) 1
C 78 2 ( 3 , 13 ) PGL ( 2 , 13 ) 1 C 310 ( 5 , 31 ) Aut ( P S L ( 5 , 2 ) ) 3
CD ( 2 p , 7 ) 7 ∣ p − 1 D 2 p × Z 7 1

Theorem 1.1

Let Γ be a connected symmetric graph of valency 7 and order 2 p q n , with p < q odd primes and n ≥ 2 . Then Γ is s -transitive with 1 ≤ s ≤ 3 and is a normal cover of one of the basic graphs Σ listed in Table 1.

For any given positive integer k , a result of Conder et al. [4] tells us that there are only finitely many connected 2-arc-transitive 7-valent graphs of order k p or k p 2 with p a prime. Theorem 1.1 together with [13, Theorem 1.1] (for case p = 2 ) has the following corollary.

Corollary 1.2

  1. For any given positive integer n , there are only finitely many connected 2-arc-transitive 7-valent graphs of order 2 p q n with 7 ≠ p < q primes.

  2. For any given positive integer n , there is no connected 2-arc-transitive 7-valent graphs of order 2 p q n with 7 < p < q primes.

2 Preliminaries

We introduce some examples and background results in this section.

2.1 Examples

As usual, for a positive integer n , denote by K n , K n , n and K n , n − n K 2 the complete graph of order n , the complete bipartite graph of order 2 n and the graph deleted a 1-matching from K n , n , respectively. Also, the Hoffman-Singleton graph of order 50 and valency 7 is denoted by HS ( 50 ) .

For a group G and a subset S ⊆ G ⧹ { 1 } , with S = S − 1 ≔ { g − 1 ∣ g ∈ S } , the Cayley graph of group G with respect to S is with vertex set G and two vertices g and h are adjacent if and only if h g − 1 ∈ S . This Cayley graph is denoted by Cay ( G , S ) .

Example 2.1

Let G = ⟨ a , b ∣ a m = b 2 = 1 , a b = a − 1 ⟩ ≅ D 2 m be a dihedral group, with m a positive integer. Let k be a solution of the congruence equation

x 6 + x 5 + ⋯ + x + 1 ≡ 0 ( mod m ) .

Define a Cayley graph

CD ( 2 m , 7 ) = Cay ( G , { b , a b , a k + 1 b , … , a k 5 + k 4 + ⋯ + 1 b } ) .

Then CD ( 2 m , 7 ) is a connected arc-transitive graph of valency 7. In particular, if m ≥ 29 , then CD ( 2 m , 7 ) is arc-regular and Aut ( CD ( 2 m , 7 ) ) = D 2 m : Z 7 , see [14, Theorem 3.1].

The following are several specific examples, see [12, Section 3].

Example 2.2

  1. There is unique connected symmetric 7-valent graph of order 30, denoted by C 30 , and Aut ( C 30 ) = S 8 ;

  2. There are exactly two connected symmetric 7-valent graphs of order 78, denoted by C 78 1 and C 78 2 , and Aut ( C 78 1 ) = PSL ( 2 , 13 ) and Aut ( C 78 2 ) = PGL ( 2 , 13 ) ;

  3. There is unique connected symmetric 7-valent graph of order 310, denoted by C 310 , and Aut ( C 310 ) = Aut ( PSL ( 5 , 2 ) ) .

Lemma 2.3

Let Γ be a connected 7-valent symmetric graph. Then the following statements hold, where p < q are primes.

  1. [2, Table 1] If ∣ V Γ ∣ = 2 p , then Γ = K 7 , 7 or CD ( 2 p , 7 ) with 7 ∣ p − 1 .

  2. [12, Section 4] If ∣ V Γ ∣ = 2 p q , then Γ = C 30 , C 78 1 , C 78 2 , C 310 or CD ( 2 p q , 7 ) with p = 7 or 7 ∣ p − 1 and 7 ∣ q − 1 .

2.2 Background results

For a positive integer m and a group T , denote by π ( m ) the number of primes which divide m , and by π ( T ) the set of primes dividing ∣ T ∣ . The group T is called a K n -group if ∣ π ( T ) ∣ = n . The simple K n -groups with 3 ≤ n ≤ 6 are classified in [15] and [16].

Theorem 2.4

[16, Theorem A] Let T be a simple K 5 -group. Then one of the following holds:

  1. T = PSL ( 2 , q ) with π ( q 2 − 1 ) = 4 ;

  2. T = PSU ( 3 , q ) with π ( ( q 2 − 1 ) ( q 3 + 1 ) ) = 4 ;

  3. T = PSL ( 3 , q ) with π ( ( q 2 − 1 ) ( q 3 − 1 ) ) = 4 ;

  4. T = O 5 ( q ) with π ( q 4 − 1 ) = 4 ;

  5. T = S z ( 2 2 m + 1 ) with π ( ( 2 2 m + 1 − 1 ) ( 2 4 m + 2 + 1 ) ) = 4 ;

  6. T = R ( 3 2 m + 1 ) with π ( ( 3 4 m + 2 − 1 ) ) = 3 and π ( 3 4 m + 2 − 3 2 m + 1 + 1 ) = 1 ;

  7. T = A 11 , A 12 , M 22 , J 3 , H S , H e , M c L , PSL ( 4 , 4 ) , PSL ( 4 , 5 ) , PSL ( 4 , 7 ) , PSL ( 5 , 2 ) , PSL ( 5 , 3 ) , PSL ( 6 , 2 ) , O 7 ( 3 ) , O 9 ( 2 ) , P S p ( 6 , 3 ) , P S p ( 8 , 2 ) , PSU ( 4 , 4 ) , PSU ( 4 , 5 ) , PSU ( 4 , 7 ) , PSU ( 4 , 9 ) , PSU ( 5 , 3 ) , PSU ( 6 , 2 ) , O + ( 8 , 3 ) , O − ( 8 , 2 ) , 3 D 4 ( 3 ) , G 2 ( 4 ) , G 2 ( 5 ) , G 2 ( 7 ) or G 2 ( 9 ) .

The vertex stabilizers of connected 7-valent symmetric graphs are known, refer to [17, Theorem 3.4], where F n with n a positive integer denotes the Frobenius group of order n .

Lemma 2.5

Let Γ be a connected 7-valent ( G , s ) -transitive graph, where G ≤ Aut Γ and s ≥ 1 . Then s ≤ 3 and one of the following holds, where α ∈ V Γ .

  1. If G α is soluble, then ∣ G α ∣ ∣ 2 2 ⋅ 3 2 ⋅ 7 . Furthermore, the couple ( s , G α ) is listed 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

  1. If G α is insoluble, then ∣ G α ∣ ∣ 2 24 ⋅ 3 4 ⋅ 5 2 ⋅ 7 . Furthermore, the couple ( s , G α ) is listed in the following table.

s 2 3
G α PSL ( 3 , 2 ) , ASL ( 3 , 2 ) , PSL ( 3 , 2 ) × S 4 , A 7 × A 6 , S 7 × S 6 , ( A 7 × A 6 ) : Z 2 ,
ASL ( 3 , 2 ) × Z 2 , A 7 , S 7 Z 2 6 : ( SL ( 2 , 2 ) × SL ( 3 , 2 ) ) , [ 2 20 ] : ( SL ( 2 , 2 ) × SL ( 3 , 2 ) )
∣ G α ∣ 2 3 ⋅ 3 ⋅ 7 , 2 6 ⋅ 3 ⋅ 7 , 2 6 ⋅ 3 2 ⋅ 7 , 2 6 ⋅ 3 4 ⋅ 5 2 ⋅ 7 , 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 7 , 2 7 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ,
2 7 ⋅ 3 ⋅ 7 , 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 , 2 4 ⋅ 3 2 ⋅ 5 ⋅ 7 2 10 ⋅ 3 2 ⋅ 7 , 2 24 ⋅ 3 2 ⋅ 7

In particular, if 5 ∣ ∣ G α ∣ , then ∣ G α ∣ ∣ 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 7 , and G α Γ ( α ) ≅ A 7 or S 7 ; if 5 ∤ ∣ G α ∣ , then ∣ G α ∣ ∣ 2 24 ⋅ 3 2 ⋅ 7 .

The following theorem is a special case of [18, Lemma 2.5], which slightly improves a nice result of Praeger [19, Theorem 4.1].

Theorem 2.6

Let Γ be a connected G -arc-transitive graph of odd prime valency, and let N ⊲ G have more than two orbits on V Γ , where G ≤ A u t Γ . Then the following statements hold.

  1. N is semiregular on V Γ , G / N ≤ A u t Γ / N , Γ / N is G / N -arc-transitive, and Γ is a normal N -cover of Γ / N ;

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

  3. G α ≅ ( G / N ) δ , for all α ∈ V Γ and δ ∈ V ( Γ / N ) .

A transitive permutation group X ≤ S y m ( Ω ) is called quasiprimitive if each minimal normal subgroup of X is transitive on Ω , while X is called biquasiprimitive if each of its minimal normal subgroups has at most two orbits and there exists one minimal normal subgroup which has exactly two orbits on Ω .

We have a next generalization of [6, Lemma 5.1].

Lemma 2.7

Let Γ be a connected G -arc-transitive r -valent graph of order 2 q n , where G ≤ Aut Γ , n ≥ 2 , and r ≥ 5 and q ≥ 5 are primes. Then either Γ = K 7 , 7 or HS ( 50 ) , or G has a minimal normal elementary abelian q -subgroup.

Proof

If G is quasiprimitive or biquasiprimitive on V Γ , by [20, Theorem 1.2], Lemma 2.7 is true. Suppose that G is neither quasiprimitive nor biquasiprimitive on V Γ . Then G has a minimal normal subgroup N , which has at least three orbits on V Γ , by Theorem 2.6, N is semiregular on V Γ and hence ∣ N ∣ ∣ 2 q n . It follows that either N = Z 2 or N = Z q d for some d < n . For the former case, the normal quotient graph Γ / N is arc-transitive of odd order q n and odd valency r , a contradiction. Therefore, N = Z q d , as required.□

3 Technical lemmas

The two lemmas regarding simple groups in this section are based on the classifications of simple K n -groups with 3 ≤ n ≤ 6 , obtained in [15] and [16].

Lemma 3.1

Let r < s be odd primes, and let T be a nonabelian simple group such that ∣ T ∣ ∣ 2 25 ⋅ 3 2 ⋅ 7 r s l and 7 r s l ∣ ∣ T ∣ for l ≥ 1 . Then one of the following holds.

  1. ∣ π ( T ) ∣ = 4 , and T is isomorphic to one of the groups listed in Table 2.

  2. ∣ π ( T ) ∣ = 5 , and T is isomorphic to one of the groups listed in Table 3. In particular, l = 1 .

Table 2

The simple K 4 -groups in Lemma 3.1

T ∣ T ∣ T ∣ T ∣ T ∣ T ∣
J 2 2 7 ⋅ 3 3 ⋅ 5 2 ⋅ 7 A 7 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 A 8 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7
PSL ( 2 , 13 ) 2 2 ⋅ 3 ⋅ 7 ⋅ 13 PSL ( 2 , 27 ) 2 2 ⋅ 3 3 ⋅ 7 ⋅ 13 PSL ( 2 , 97 ) 2 5 ⋅ 3 ⋅ 7 2 ⋅ 97
PSL ( 2 , 127 ) 2 3 ⋅ 3 2 ⋅ 7 ⋅ 127 PSL ( 3 , 4 ) 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 PSL ( 3 , 8 ) 2 9 ⋅ 3 2 ⋅ 7 2 ⋅ 73
PSU ( 3 , 5 ) 2 4 ⋅ 3 2 ⋅ 5 3 ⋅ 7
Table 3

The simple K 5 -groups in Lemma 3.1

T M 22 PSL ( 5 , 2 )
∣ T ∣ 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 2 10 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 31
T PSL ( 2 , 2 6 ) PSL ( 2 , 29 )
∣ T ∣ 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 13 2 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 29
T PSL ( 2 , 41 ) PSL ( 2 , 43 )
∣ T ∣ 2 3 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 41 2 10 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 31
T PSL ( 2 , 71 ) PSL ( 2 , 83 )
∣ T ∣ 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 71 2 2 ⋅ 3 ⋅ 7 ⋅ 41 ⋅ 83
T PSL ( 2 , 113 ) PSL ( 2 , 167 )
∣ T ∣ 2 4 ⋅ 3 ⋅ 7 ⋅ 19 ⋅ 113 2 3 ⋅ 3 ⋅ 7 ⋅ 83 ⋅ 167
T PSL ( 2 , 223 ) PSL ( 2 , 503 )
∣ T ∣ 2 5 ⋅ 3 ⋅ 7 ⋅ 37 ⋅ 223 2 3 ⋅ 3 2 ⋅ 7 ⋅ 251 ⋅ 503
T PSL ( 2 , 673 ) PSL ( 2 , 2017 )
∣ T ∣ 2 5 ⋅ 3 ⋅ 7 ⋅ 337 ⋅ 673 2 5 ⋅ 3 2 ⋅ 7 ⋅ 1009 ⋅ 2017
T PSL ( 2 , 3583 ) PSL ( 2 , 64513 )
∣ T ∣ 2 9 ⋅ 3 2 ⋅ 7 ⋅ 199 ⋅ 3583 2 10 ⋅ 3 2 ⋅ 7 ⋅ 32257 ⋅ 64513
T PSL ( 2 , 2752513 ) PSL ( 2 , 16515073 )
∣ T ∣ 2 17 ⋅ 3 ⋅ 7 ⋅ 1376257 ⋅ 2752513 2 18 ⋅ 3 2 ⋅ 7 ⋅ 8257537 ⋅ 16515073

Proof

Clearly, 3 ≤ ∣ π ( T ) ∣ ≤ 5 . If ∣ π ( T ) ∣ = 3 , by [15, Theorem I], there are exactly eight specific simple K 3 -groups listed in [15, Table 1], checking the orders, no group T exists in this case.

  1. Suppose ∣ π ( T ) ∣ = 4 . By [15, Theorem I], either

    1. T is isomorphic to one of the groups listed in [15, Table 2]; or

    2. T = PSL ( 2 , q ) for some prime power q .

Assume (a) occurs. Suppose 5 ∈ π ( T ) . As 7 ∈ π ( T ) , by checking [15, Table 2], T is a { 2 , 3 , 5 , 7 } -group, and so r , s ∈ { 3 , 5 , 7 } . If s = 7 , then r = 3 or 5, and one easily checks that no group T exists in the case. If s = 5 , then r = 3 , so ∣ T ∣ ∣ 2 25 ⋅ 3 3 ⋅ 5 l ⋅ 7 and 7 ⋅ 3 ⋅ 5 l ∣ ∣ T ∣ , and one may derive that T = J 2 , A 7 , A 8 , PSL ( 3 , 4 ) and PSU ( 3 , 5 ) . Suppose now 5 ∉ π ( T ) . By [15, Table 2], r = 3 or 7 and s > 7 , hence ∣ T ∣ ∣ 2 25 ⋅ 3 3 ⋅ 7 2 ⋅ s l , by checking the orders, we obtain T = PSL ( 3 , 8 ) .

Now assume (b) occurs. If q is a power of 2 , 3 or 7, by [15, Table 3], the only example is T = PSL ( 2 , 27 ) . For the other cases, by [15, Theorem 3.2], q ≥ 11 is a prime, note that ∣ PSL ( 2 , q ) ∣ is always divisible by 3, we conclude that T is a { 2 , 3 , 7 , q } -group, hence s = q , l = 1 , and r ∈ { 3 , 7 } . Furthermore, since ∣ T ∣ ∣ 2 25 ⋅ 3 2 ⋅ 7 r s l , we have q − 1 2 ⋅ q + 1 2 ∣ 2 24 ⋅ 3 3 ⋅ 7 2 , and as q − 1 2 , q + 1 2 = 1 , it follows that either q + 1 ∣ 2 ⋅ 3 3 ⋅ 7 2 if 2 ∣ q − 1 2 or q − 1 ∣ 2 ⋅ 3 3 ⋅ 7 2 if 2 ∣ q + 1 2 . Then a computation by [21] shows q ∈ { 13 , 17 , 19 , 41 , 43 , 53 , 97 , 127 , 293 , 379 , 881 , 883 } . Checking the orders, we obtain T = PSL ( 2 , 13 ) , PSL ( 2 , 97 ) or PSL ( 2 , 127 ) .

  1. Suppose ∣ π ( T ) ∣ = 5 . Then T is a { 2 , 3 , 7 , r , s } -group and satisfies part (a)–(g) of Theorem 2.4. We analyze these cases one by one in the following. Note that ∣ T ∣ ∣ 2 25 ⋅ 3 2 ⋅ 7 r s l , we obtain

(1) 2 26 ∤ ∣ T ∣ , 3 3 ∤ ∣ T ∣ , 7 2 ∤ ∣ T ∣ , r 2 ∤ ∣ T ∣ .

Assume part (a) of Theorem 2.4 occurs. Then

(2) ∣ T ∣ = ∣ PSL ( 2 , q ) ∣ = 1 ( 2 , q − 1 ) q ( q − 1 ) ( q + 1 ) .

Since ∣ T ∣ ∣ 2 25 ⋅ 3 2 ⋅ 7 r s l and 7 r s l ∣ ∣ T ∣ , we have q ≠ 3 , 3 2 , 7 or r , and then derive from equation (1) that q = 2 i with 1 ≤ i ≤ 25 or q = s l . For the former case, since π ( q 2 − 1 ) = 4 , we get q = 2 6 , 2 8 , 2 9 , 2 11 or 2 23 , and by checking the orders, we obtain T = PSL ( 2 , 2 6 ) . For the latter case, we have q + 1 2 ⋅ q − 1 2 ∣ 2 24 ⋅ 3 2 ⋅ 7 r , and as q + 1 2 , q − 1 2 = 1 , it follows that either q − 1 ∣ 2 25 ⋅ 3 2 ⋅ 7 if r ∣ q + 1 2 , or q + 1 ∣ 2 25 ⋅ 3 2 ⋅ 7 if r ∣ q − 1 2 . Recall that π ( q 2 − 1 ) = 4 , q = s l with s > 7 and T satisfies equation (1), a direct computation by [21] shows that q = 29 , 41 , 43 , 71 , 83 , 113 , 167 , 223 , 503 , 673 , 2017 , 3583 , 64513 , 2752513 or 16515073, as in Table 3.

Assume part (b) occurs. Then

(3) ∣ T ∣ = ∣ PSU ( 3 , q ) ∣ = 1 ( 3 , q + 1 ) q 3 ( q − 1 ) ( q + 1 ) 2 ( q 2 − q + 1 ) .

By equation (1), q is a 2-power or an s -power. If q is a 2-power, then q = 2 i with 1 ≤ i ≤ 8 , and as π ( ( q 2 − 1 ) ( q 3 + 1 ) ) = 4 , we get q = 2 4 , 2 5 or 2 7 ; however, in these three cases, ∣ T ∣ is always not divisible by 7, a contradiction. If q is an s -power, as 7 2 ∤ ∣ T ∣ and r 2 ∤ ∣ T ∣ , we obtain ( q + 1 ) 2 ∣ 2 25 ⋅ 3 2 , or equivalently q + 1 ∣ 2 12 ⋅ 3 . Since π ( ( q 2 − 1 ) ( q 3 + 1 ) ) = 4 , computation in [21] shows q = 11 and 23; however, in both cases, ∣ T ∣ is not divisible by 7, also a contradiction.

Assume part (c) occurs. Then

(4) ∣ T ∣ = ∣ PSL ( 3 , q ) ∣ = 1 ( 3 , q − 1 ) q 3 ( q − 1 ) 2 ( q + 1 ) ( q 2 + q + 1 ) .

By equation (1), one derives q is a 2-power or an s -power. Then with similar discussion to that in part (b) above, one may draw a contradiction.

Assume part (d) occurs. Then

(5) ∣ T ∣ = ∣ O 5 ( q ) ∣ = 1 2 q 4 ( q 4 − 1 ) ( q 3 − 1 ) ( q 2 − 1 ) .

By equation (1), q is a 2-power or an s -power. If q is a 2-power, then q = 2 i with 1 ≤ i ≤ 6 . Since π ( q 4 − 1 ) = 4 , we have q = 2 3 or 2 4 , and ∣ T ∣ = 2 12 ⋅ 3 4 ⋅ 5 ⋅ 7 2 ⋅ 13 or 2 16 ⋅ 3 2 ⋅ 5 2 ⋅ 1 7 2 ⋅ 257 , respectively, contradicting equation (1). If q is an s -power, as 7 2 ∤ ∣ T ∣ and r 2 ∤ ∣ T ∣ , we have ( q 2 − 1 ) 2 ∣ 2 25 ⋅ 3 2 , and so q + 1 2 ⋅ q − 1 2 ∣ 2 11 ⋅ 3 . Noting that q + 1 2 , q − 1 2 = 1 , we conclude that either q + 1 2 ∣ 3 or q − 1 2 ∣ 3 , implying s ≤ q ≤ 7 , a contradiction.

Assume part (e) occurs. Then ∣ T ∣ = ∣ S z ( 2 2 m + 1 ) ∣ = 2 4 m + 2 ( 2 4 m + 2 + 1 ) ( 2 2 m + 1 − 1 ) . Since 2 26 ∤ ∣ T ∣ and π ( ( 2 2 m + 1 − 1 ) ( 2 4 m + 2 + 1 ) ) = 4 , we derive m = 3 ; however, ∣ S z ( 2 7 ) ∣ = 2 14 ⋅ 5 ⋅ 29 ⋅ 113 ⋅ 127 is not divisible by 7, a contradiction.

Assume part (f) occurs. Then ∣ T ∣ = ∣ R ( 3 2 m + 1 ) ∣ = 3 6 m + 3 ( 3 6 m + 3 + 1 ) ( 3 2 m + 1 − 1 ) , so 3 9 ∣ ∣ T ∣ , contradicting 3 3 ∤ ∣ T ∣ .

Finally, assume part (g) occurs. Checking the orders of the 30 specific simple groups there, we obtain T = M 22 and P S L ( 5 , 2 ) .□

Lemma 3.2

Let r < s be odd primes, and let T be a nonabelian simple group such that ∣ T ∣ ∣ 2 9 ⋅ 3 4 ⋅ 5 2 ⋅ 7 r s l and 35 r s l ∣ ∣ T ∣ with l ≥ 1 . Then one of the following holds.

  1. ∣ π ( T ) ∣ = 4 , and T is isomorphic to one of the groups listed in Table 4.

  2. ∣ π ( T ) ∣ = 5 , l = 1 and T is isomorphic to one of the groups listed in Table 5.

  3. ∣ π ( T ) ∣ = 6 , l = 1 and T = J 1 , M 23 , or PSL ( 2 , q ) with q = 139 , 181, 211, 239, 281, 349, 379, 421, 601, 631, 701, 769, 811, 839, 1009, 1049, 1051, 1399, 1511, 1889, 2099, 2239, 2267, 2269, 2591, 2689, 2801, 3779, 4481, 6481, 6719, 7559, 10079, 12601, 15121, 21601, 26881, 28351, 30241, 37799, 53759, 56701, 69119, 96769, 172801, 201599, 453599, 483839 or 907199.

Table 4

The simple K 4 -groups in Lemma 3.2

T ∣ T ∣ T ∣ T ∣ T ∣ T ∣
J 2 2 7 ⋅ 3 3 ⋅ 5 2 ⋅ 7 A 10 2 7 ⋅ 3 4 ⋅ 5 2 ⋅ 7 PSU ( 3 , 5 ) 2 4 ⋅ 3 2 ⋅ 5 3 ⋅ 7
PSp ( 4 , 7 ) 2 8 ⋅ 3 2 ⋅ 5 2 ⋅ 7 4 PSL ( 2 , 49 ) 2 4 ⋅ 3 ⋅ 5 2 ⋅ 7 2
Table 5

The simple K 5 -groups in Lemma 3.2

T ∣ T ∣ T ∣ T ∣
A 11 2 7 ⋅ 3 4 ⋅ 5 2 ⋅ 7 ⋅ 11 A 12 2 9 ⋅ 3 5 ⋅ 5 2 ⋅ 7 ⋅ 11
M 22 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 HS 2 9 ⋅ 3 2 ⋅ 5 3 ⋅ 7 ⋅ 11
PSL ( 2 , 2 6 ) 2 6 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 13 PSL ( 2 , 5 3 ) 2 3 ⋅ 3 2 ⋅ 5 3 ⋅ 7 ⋅ 31
PSL ( 2 , 29 ) 2 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 29 PSL ( 2 , 41 ) 2 3 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 41
PSL ( 2 , 71 ) 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 71 PSL ( 2 , 251 ) 2 2 ⋅ 3 2 ⋅ 5 3 ⋅ 7 ⋅ 251
PSL ( 2 , 449 ) 2 6 ⋅ 3 2 ⋅ 5 2 ⋅ 7 ⋅ 449

Proof

Obviously, 3 ≤ ∣ π ( T ) ∣ ≤ 6 . If ∣ π ( T ) ∣ = 3 , by [15, Theorem I], T is isomorphic to one of the eight groups listed in [15, Table 1]. However, the order of each group is not divisible by 35, a contradiction.

  1. Assume ∣ π ( T ) ∣ = 4 . By [15, Theorem I], T is isomorphic to one of the groups listed in [15, Table 3] or T = PSL ( 2 , q ) for some prime power q . For the former case, since 35 r s l ∣ ∣ T ∣ and ∣ T ∣ ∣ 2 9 ⋅ 3 4 ⋅ 5 2 ⋅ 7 r s l , one easily derives that T = J 2 , A 10 , PSU ( 3 , 5 ) or PSP ( 4 , 7 ) . For the latter case, note that 3 ∣ ∣ PSL ( 2 , q ) ∣ and 35 ∣ ∣ T ∣ , T is a { 2 , 3 , 5 , 7 } -group, then by [15, Table 3], the only example is T = PSL ( 2 , 49 ) .

  2. Assume ∣ π ( T ) ∣ = 5 . Then s > 7 and T satisfies parts (a)–(g) of Theorem 2.4. Since ∣ T ∣ ∣ 2 9 ⋅ 3 4 ⋅ 5 2 ⋅ 7 r s l , we have

    (6) 2 10 ∤ ∣ T ∣ , q 3 6 ∤ ∣ T ∣ , 5 4 ∤ ∣ T ∣ , 7 3 ∤ ∣ T ∣ .

Suppose T = PSL ( 2 , q ) , as in part (a) of Theorem 2.4. Then T is a { 2 , 3 , 5 , 7 , s } -group as 3 ∣ ∣ PSL ( 2 , q ) ∣ . If q is a 2-power, then q = 2 6 , 2 8 or 2 9 since 2 10 ∤ ∣ T ∣ and π ( q 2 − 1 ) = 4 , by checking the orders, we obtain T = PSL ( 2 , 2 6 ) . If q is a 3-power, then q ∈ { 3 , 3 2 , 3 3 , 3 4 , 3 5 } since 3 6 ∤ ∣ T ∣ , it follows that π ( q 2 − 1 ) ≠ 4 , a contradiction. If q is a 5-power, then q = 5 3 because 5 4 ∤ ∣ T ∣ and π ( q 2 − 1 ) = 4 , which gives rise to an example T = PSL ( 2 , 5 3 ) . If q is a 7-power, then q = 7 or 7 2 since 7 3 ∤ ∣ T ∣ , contradicting π ( q 2 − 1 ) = 4 . Now, assume that q is an s -power. Then q + 1 2 ⋅ q − 1 2 ∣ 2 9 ⋅ 3 5 ⋅ 5 3 ⋅ 7 2 . Since q + 1 2 , q − 1 2 = 1 , we have q − 1 2 ∣ 3 5 ⋅ 5 3 ⋅ 7 2 or q + 1 2 ∣ 3 5 ⋅ 5 3 ⋅ 7 2 . Recall that π ( q 2 − 1 ) = 4 , computation in [21] shows q ∈ { 29, 41, 43, 71, 89, 149, 151, 251, 269, 271, 293, 449, 751, 809, 2251, 2647, 4051, 7937, 12149, 20249, 23813 } . Checking the orders, we obtain the examples T = PSL ( 2 , 29 ) , PSL ( 2 , 41 ) and PSL ( 2 , 449 ) .

Suppose T = PSU ( 3 , q ) , as in part (b). Since π ( ( q 2 − 1 ) ( q 3 + 1 ) ) = 4 , by equations (3) and (6), we derive that q is an s -power, and ( q + 1 ) 2 ∣ 2 10 ⋅ 3 5 ⋅ 5 3 ⋅ 7 2 , so q + 1 ∣ 2 5 ⋅ 3 2 ⋅ 5 ⋅ 7 . Since π ( ( q 2 − 1 ) ( q 3 + 1 ) ) = 4 , a computation by [21] shows that q ∈ { 11 , 13 , 17 , 19 , 23 } ; however, by checking the orders, no group T exists in the case. Similarly, one may exclude part (c), namely T = PSL ( 3 , q ) .

Suppose T = S z ( 2 2 m + 1 ) or R ( 3 2 m + 1 ) , as in part (d) or (e). Then ∣ T ∣ = 2 4 m + 2 ( 2 4 m + 2 + 1 ) ( 2 2 m + 1 − 1 ) or 3 6 m + 3 ( 3 6 m + 3 + 1 ) ( 3 2 m + 1 − 1 ) , respectively. Since ∣ π ( S z ( 2 3 ) ) ∣ = 4 , T ≠ S z ( 2 3 ) , and hence 2 10 ∣ ∣ T ∣ or 3 9 ∣ ∣ T ∣ , contradicting equation (6).

Suppose T = O 5 ( q ) , as in part (f). Since π ( q 4 − 1 ) = 4 , by equations (5) and (6), we conclude that q is an s -power, and ( q − 1 ) 3 ∣ 2 10 ⋅ 3 5 ⋅ 5 3 , hence q − 1 ∣ 2 3 ⋅ 3 ⋅ 5 . It follows q = 11 and ∣ T ∣ = ∣ O 5 ( 11 ) ∣ = 2 8 ⋅ 3 2 ⋅ 5 2 ⋅ 1 1 4 ⋅ 61 is not divisible by 7, a contradiction.

Finally, suppose T lies in the groups listed in part (g). Checking the orders, we obtain T = A 11 , A 12 , M 22 , or HS .

  1. Assume that ∣ π ( T ) ∣ = 6 . Then 7 < r < s and s > 11 . By [16, Theorem B], one of the following holds:

    1. T = PSL ( 2 , q ) where π ( q 2 − 1 ) = 5 ;

    2. T = PSL ( 3 , q ) where π ( ( q 2 − 1 ) ( q 3 − 1 ) ) = 5 ;

    3. T = PSL ( 4 , q ) where π ( ( q 2 − 1 ) ( q 3 − 1 ) ( q 4 − 1 ) ) = 5 ;

    4. T = PSU ( 3 , q ) where π ( ( q 2 − 1 ) ( q 3 + 1 ) ) = 5 ;

    5. T = PSU ( 4 , q ) where π ( ( q 2 − 1 ) ( q 3 + 1 ) ( q 4 − 1 ) ) = 5 ;

    6. T = O 5 ( q ) where π ( q 4 − 1 ) = 5 ;

    7. T = G 2 ( q ) where π ( q 6 − 1 ) = 5 ;

    8. T = S z ( 2 2 m + 1 ) where π ( ( 2 2 m + 1 − 1 ) ( 2 4 m + 2 + 1 ) ) = 5 ;

    9. T = R ( 3 2 m + 1 ) where π ( ( 3 2 m + 1 − 1 ) ( 3 6 m + 3 + 1 ) ) = 5 ;

    10. T is one of the 38 groups listed in [16, Theorem B].

Recall that ∣ T ∣ ∣ 2 9 ⋅ 3 4 ⋅ 5 2 ⋅ 7 r s l , then we have

(7) 2 10 ∤ ∣ T ∣ , 3 5 ∤ ∣ T ∣ , 5 3 ∤ ∣ T ∣ , 7 2 ∤ ∣ T ∣ , r 2 ∤ ∣ T ∣ .

Since ∣ S z ( 2 2 m + 1 ) ∣ = 2 4 m + 2 ( 2 4 m + 2 + 1 ) ( 2 2 m + 1 − 1 ) and 2 10 ∤ ∣ T ∣ , we have m = 1 and ∣ π ( T ) ∣ = ∣ π ( S z ( 8 ) ) ∣ = 4 ≠ 6 , this contradiction excludes case (h). Since ∣ R ( 3 2 m + 1 ) ∣ = 3 6 m + 3 ( 3 6 m + 3 + 1 ) ( 3 2 m + 1 − 1 ) , 3 9 ∣ ∣ T ∣ , contradicting equation (7), this excludes case (i). For case (j), by checking the orders, we have T = J 1 or M 23 .

Suppose case (a) occurs. Since π ( q 2 − 1 ) = 5 , q ≠ 3 , 3 2 , 3 3 , 3 4 , 5 , 5 2 , 7 , and by equation (7), we have that either q = 2 i ( 1 ≤ i ≤ 9 ) or s l . The former case does not give examples by checking the orders. For the latter case, by equation (2), we have q − 1 2 ⋅ q + 1 2 ∣ 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 7 r , and as ( q − 1 2 , q + 1 2 ) = 1 , it follows that either q + 1 2 ∣ 2 9 ⋅ 3 4 ⋅ 5 2 ⋅ 7 or q − 1 2 ∣ 2 9 ⋅ 3 4 ⋅ 5 2 ⋅ 7 . Recall that π ( q 2 − 1 ) = 5 and s > 11 , computation in [21] shows that q lies in part (iii) of Lemma 3.2.

Suppose case (b) occurs. Since π ( ( q 2 − 1 ) ( q 3 − 1 ) ) = 5 , q ≠ 2 , 2 2 , 2 3 or 3, and by equations (4) and (7), we derive that q is an s -power, and ( q − 1 ) 2 ∣ 2 9 ⋅ 3 4 ⋅ 5 2 , implying q − 1 ∣ 2 4 ⋅ 3 2 ⋅ 5 . Since π ( ( q 2 − 1 ) ( q 3 − 1 ) ) = 5 , a computation by [21] shows that q = 37 , 41 or 241, which does not give rise to examples by checking the orders. Similarly, one may exclude case (d).

Suppose case (e) occurs. Then

∣ T ∣ = ∣ PSU ( 4 , q ) ∣ = 1 ( 4 , q + 1 ) q 6 ( q 2 − 1 ) 2 ( q 2 + 1 ) ( q 3 + 1 ) .

By equation (7), one may conclude that q is an s -power, and ( q 2 − 1 ) 2 ∣ 2 9 ⋅ 3 4 ⋅ 5 2 , hence q − 1 2 ⋅ q + 1 2 ∣ 2 2 ⋅ 3 2 ⋅ 5 . As q − 1 2 , q + 1 2 = 1 , we obtain q − 1 2 ∣ 2 2 ⋅ 5 or q + 1 2 ∣ 2 2 ⋅ 5 . It follows that q = 19 since π ( ( q 2 − 1 ) ( q 3 + 1 ) ( q 4 − 1 ) ) = 5 , and ∣ T ∣ = ∣ PSU ( 4 , 19 ) ∣ = 2 7 ⋅ 3 4 ⋅ 5 3 ⋅ 7 3 ⋅ 1 9 6 ⋅ 181 , contradiction equation (7). For case (c), then

∣ T ∣ = ∣ PSL ( 4 , q ) ∣ = 1 ( 4 , q − 1 ) q 6 ( q 2 − 1 ) ( q 3 − 1 ) ( q 4 − 1 ) ,

a similar argument to that in case (e) may draw a contradiction.

Suppose case (f) occurs. Since π ( q 4 − 1 ) = 5 , q ≠ 2 , 2 2 and 3, by equations (5) and (7), we conclude that q is an s -power, and ( q 2 + 1 ) ( q 3 − 1 ) ( q 2 − 1 ) 2 ∣ 2 10 ⋅ 3 4 ⋅ 5 2 ⋅ 7 r , hence ( q 2 − 1 ) 2 ∣ 2 10 ⋅ 3 4 ⋅ 5 2 , or equivalently q 2 − 1 ∣ 2 5 ⋅ 3 2 ⋅ 5 . As discussed in the above paragraph, one may derive that q = 17 , 19 , 29 , 31 or 89, which does not give rise to example by checking the orders. Finally, for case (g), ∣ T ∣ = ∣ G 2 ( q ) ∣ = q 6 ( q 6 − 1 ) ( q 2 − 1 ) , a similar discussion may draw a contradiction.□

4 Vertex quasiprimitive and vertex biquasiprimitive cases

Let Γ be a connected G -arc-transitive 7-valent graph of order 2 p q n , where G ≤ Aut Γ , p < q are odd primes and n ≥ 2 . Let N be a minimal normal subgroup of G . Then N = T d , with T a simple group and d ≥ 1 . Let α ∈ V Γ .

Lemma 4.1

If N is nonabelian, then d = 1 .

Proof

Suppose for a contradiction that N is nonabelian and d ≥ 2 . Then ∣ N ∣ ∤ 2 p q n , N α ≠ 1 and N has at most two orbits on V Γ by Theorem 2.6. Set N = T 1 × T 2 × ⋯ × T d with each T i ≅ T .

Assume first N is transitive on V Γ . Since 1 ≠ N α ⊲ G α and Γ is connected, we have 1 ≠ N α Γ ( α ) ⊲ G α Γ ( α ) . It follows that N α Γ ( α ) is transitive, and Γ is N -arc-transitive. If T 1 is transitive on V Γ , then the centralizer C N ( T 1 ) is semiregular on V Γ (see [22, Theorem 4.2A]), so is T 2 , which is a contradiction as ∣ T 2 ∣ does not divide ∣ V Γ ∣ = 2 p q n ; if T 1 has at least three orbits on V Γ , by Theorem 2.6, T 1 is semiregular, again a contradiction. Therefore, T 1 has exactly two orbits, say U and W , on V Γ . Since T 1 ⊲ N , U and W form an N -block system on V Γ . It follows that the set stabilizer N U is of index 2 in N , which is a contradiction because N = T d has no subgroup with index 2.

Assume now N has exactly two orbits, say Δ 1 and Δ 2 , on V Γ . Then Γ is a bipartite graph with bipartitions Δ 1 and Δ 2 . Let G + = G Δ 1 = G Δ 2 , the stabilizer on the bipartitions. If G + acts unfaithfully on Δ 1 , by [23, Lemma 5.2], Γ is a complete bipartite graph, so Γ = K 7 , 7 as v a l ( Γ ) = 7 and hence ∣ V Γ ∣ = 14 , a contradiction. Suppose G + acts faithfully on Δ 1 . Then N ≤ G + can be viewed as a transitive permutation group on Δ 1 . If T 1 is transitive on Δ 1 , then [22, Theorem 4.2A] implies T 2 is semiregular on Δ 1 , hence ∣ T 2 ∣ ∣ p q n , a contradiction. Thus, T 1 has at least two orbits on Δ 1 . It then follows from [7, Lemma 3.2] that T 1 is semiregular on Δ 1 , also a contradiction.□

The next two lemmas exclude the vertex quasiprimitive and vertex biquasiprimitive cases.

Lemma 4.2

If G is quasiprimitive on V Γ , then no graph Γ exists.

Proof

Since G is quasiprimitive on V Γ , N is transitive on V Γ . If N is abelian, then N is regular on V Γ and so ∣ T ∣ d = ∣ N ∣ = 2 p q n , a contradiction. Thus, N is nonabelian, and then by Lemma 4.1, we have d = 1 and N = T . Furthermore, since T α ≠ 1 , we conclude that Γ is T -arc-transitive, and hence T α satisfies Lemma 2.5. We divide our proof into two cases depending on whether 5 divides ∣ T α ∣ or not.

Case 1

Assume 5 ∤ ∣ T α ∣ .

By Lemma 2.5, ∣ T α ∣ ∣ 2 24 ⋅ 3 2 ⋅ 7 , and by the transitivity of T , we have ∣ T ∣ = ∣ V Γ ∣ ∣ T α ∣ divides 2 25 ⋅ 3 2 ⋅ 7 p q n ; on the other hand, since Γ is T -arc-transitive, we have 7 ∣ ∣ T α ∣ , and so 7 p q n ∣ ∣ T ∣ . Therefore, T satisfies Lemma 3.1; in particular, ∣ π ( T ) ∣ = 4 or 5. If ∣ π ( T ) ∣ = 5 , by Lemma 3.1(ii), n = 1 , a contradiction.

Suppose ∣ π ( T ) ∣ = 4 . Noting that n ≥ 2 , by Lemma 3.1(i), we easily conclude that the triple ( T , p , q n ) = ( J 2 , 3 , 5 2 ) or ( PSU ( 3 , 5 ) , 3 , 5 3 ) . For the former case, ∣ V Γ ∣ = 2 ⋅ 3 ⋅ 5 2 , so ∣ T α ∣ = ∣ T ∣ ∣ V Γ ∣ = 2 6 ⋅ 3 2 ⋅ 7 ; however, by [24], J 2 has no subgroup with order 2 6 ⋅ 3 2 ⋅ 7 , a contradiction. For the latter case, ∣ V Γ ∣ = 2 ⋅ 3 ⋅ 5 3 , hence ∣ T α ∣ = ∣ T ∣ ∣ V Γ ∣ = 2 3 ⋅ 3 ⋅ 7 , then by Lemma 2.5, we obtain T α = PSL ( 3 , 2 ) ; however, computation in [21] shows that no graph Γ exists in this case.

Case 2

Assume 5 ∣ ∣ T α ∣ .

By Lemma 2.5, ∣ T α ∣ ∣ 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 7 , and T α Γ ( α ) ≅ A 7 or S 7 . It follows that ∣ T ∣ = ∣ V Γ ∣ ∣ T α ∣ divides 2 9 ⋅ 3 4 ⋅ 5 2 ⋅ 7 p q n ; moreover, as 35 ∣ ∣ T α ∣ , we have 35 p q n divides ∣ T ∣ . Therefore, T satisfies Lemma 3.2; in particular 4 ≤ π ( T ) ≤ 6 . If ∣ π ( T ) ∣ = 5 or 6, by Lemma 3.2, we have n = 1 , a contradiction.

Suppose ∣ π ( T ) ∣ = 4 . Since n ≥ 2 , by Lemma 3.2(i), the only possibility is T = PSp ( 4 , 7 ) and ( p , q ) = ( 3 , 7 3 ) or ( 5 , 7 3 ) . Consequently, ∣ V Γ ∣ = 2 ⋅ 3 ⋅ 7 3 or 2 ⋅ 5 ⋅ 7 3 , and ∣ T α ∣ = ∣ T ∣ ∣ V Γ ∣ = 2 7 ⋅ 3 ⋅ 5 2 ⋅ 7 or 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 , respectively. By Lemma 2.5, it is a contradiction.□

Lemma 4.3

If G is biquasiprimitive on V Γ , then no graph Γ exists.

Proof

Since G is biquasiprimitive on V Γ , G has a minimal normal subgroup N = T d which has exactly two orbits (say Δ 1 and Δ 2 ) on V Γ . Then Γ is a bipartite graph with bipartition Δ 1 and Δ 2 . Let G + = G Δ 1 = G Δ 2 . Then N ≤ G + , ∣ G : G + ∣ = 2 and G α = G α + . If N is abelian, then N is regular on Δ 1 and so ∣ T ∣ d = ∣ N ∣ = p q n , a contradiction. Hence, N is nonabelian, and by Lemma 4.1, we further conclude that N = T is a nonabelian simple group.

If G + acts unfaithfully on Δ 1 or Δ 2 , by [23, Lemma 5.2], Γ is a complete bipartite graph, so Γ = K 7 , 7 and ∣ V Γ ∣ = 14 , a contradiction.

Assume now G + acts faithfully on Δ 1 and Δ 2 . Then by [25, Theorem 1.5], either

  1. G + is quasiprimitive on Δ i ; or

  2. G + has two normal subgroups M 1 and M 2 such that M 1 ≅ M 2 are semiregular on V Γ . Furthermore, the group M 1 × M 2 is regular on Δ i .

For case (2), we have ∣ M 1 ∣ 2 = ∣ Δ i ∣ = p q n , a contradiction.

Suppose case (1) occurs. Since G + is quasiprimitive on Δ i and has a simple minimal normal subgroup T , by O’Nan-Scott-Praeger theorem ([19]), s o c ( G + ) = T or T 2 . For the latter case, G + is of holomorph type and T is regular on Δ i , so ∣ T ∣ = p q n , a contradiction. Therefore, s o c ( G + ) = T . Furthermore, if T is not the unique minimal normal subgroup of G , since G = G + . Z 2 , one easily derives G = G + × Z 2 , hence the normal subgroup Z 2 has p q n orbits on V Γ , contradicting the biquasiprimitivity of G . Thus, G is almost simple with socle T , and we may set G = T . o , and G + = T . o ′ with Z 2 ≤ o ≤ O u t ( T ) and ∣ o : o ′ ∣ = 2 .

Case 1

Assume 5 ∤ ∣ T α ∣ .

Since T α ≤ G α , by Lemma 2.5, ∣ T α ∣ ∣ 2 24 ⋅ 3 2 ⋅ 7 , and hence ∣ T ∣ = ∣ Δ 1 ∣ ∣ T α ∣ divides 2 24 ⋅ 3 2 ⋅ 7 p q n ; on the other hand, noting that T α ≠ 1 , we obtain 7 ∣ ∣ T α ∣ , and so 7 p q n ∣ ∣ T ∣ . Therefore, T satisfies Lemma 3.1, and π ( T ) = 4 or 5.

If π ( T ) = 5 , by Lemma 3.1(ii), we have n = 1 , a contradiction.

Suppose now ∣ π ( T ) ∣ = 4 . Then by Lemma 3.1(i) and note that n ≥ 2 , we have ( T , p , q n ) = ( J 2 , 3 , 5 2 ) or ( PSU ( 3 , 5 ) , 3 , 5 3 ) . For the former case, as O u t ( J 2 ) ≅ Z 2 , we have o = Z 2 , o ′ = 1 and G + = T . It follows ∣ G α ∣ = ∣ G α + ∣ = ∣ T α ∣ = ∣ T ∣ ∣ Δ 1 ∣ = 2 7 ⋅ 3 2 ⋅ 7 , which is a contradiction by Lemma 2.5. For the latter case, ∣ T α ∣ = ∣ T ∣ ∣ Δ 1 ∣ = 2 4 ⋅ 3 ⋅ 5 ⋅ 7 . Since O u t ( PSU ( 3 , 5 ) ) ≅ S 3 , we have that either o = S 3 and o ′ = Z 3 , or o = Z 2 and o ′ = 1 . Thus, ∣ G α ∣ = ∣ G α + ∣ = ∣ T α ∣ ⋅ ∣ o ′ ∣ = 2 4 ⋅ 3 ⋅ 5 ⋅ 7 or 2 4 ⋅ 3 2 ⋅ 5 ⋅ 7 . By Lemma 2.5, the only possibility is ∣ G α ∣ = 2 4 ⋅ 3 2 ⋅ 5 ⋅ 7 , and G α ≅ S 7 ; however, computation in [21] shows that no graph Γ exists in the case.

Case 2

Assume 5 ∣ ∣ T α ∣ .

As T α ⊲ G α , by Lemma 2.5, ∣ T α ∣ ∣ 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 7 , and T α Γ ( α ) ≅ A 7 or S 7 . It follows that ∣ T ∣ = ∣ V Γ ∣ ∣ T α ∣ divides 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 7 p q n ; moreover, as 35 ∣ ∣ T α ∣ , we have 35 p q n divides ∣ T ∣ . Hence, T satisfies Lemma 3.2.

If ∣ π ( T ) ∣ = 5 or 6, by Lemma 3.2, n = 1 , a contradiction.

Suppose ∣ π ( T ) ∣ = 4 . By Lemma 3.2(i), we have T = PSp ( 4 , 7 ) , and ( p , q n ) = ( 3 , 7 3 ) or ( 5 , 7 3 ) , so ∣ T α ∣ = ∣ T ∣ ∣ Δ 1 ∣ = 2 8 ⋅ 3 ⋅ 5 2 ⋅ 7 or 2 8 ⋅ 3 2 ⋅ 5 2 ⋅ 7 , respectively. Furthermore, as O u t ( P S p ( 4 , 7 ) ) ≅ Z 2 , we have o = Z 2 , o ′ = 1 and G + = T . o ′ = T . Thus, ∣ G α ∣ = ∣ G α + ∣ = ∣ T α ∣ = 2 8 ⋅ 3 ⋅ 5 2 ⋅ 7 or 2 8 ⋅ 3 2 ⋅ 5 2 ⋅ 7 , by Lemma 2.5, which is a contradiction.□

5 Proofs of Theorem 1.1

We will complete the proof of Theorem 1.1 in this final section.

Lemma 5.1

Let p < q be odd primes, and let Γ be a connected G -arc-transitive 7-valent graph of order 2 p q m , where G ≤ A u t Γ and m ≥ 2 . Then either Γ is a Z 3 -cover of HS ( 50 ) or G has a normal elementary abelian q -subgroup.

Proof

By Lemmas 4.2 and 4.3, G is neither quasiprimitive nor biquasiprimitive on V Γ . Hence, G has a minimal normal subgroup, say N , which has at least three orbits on V Γ . By Theorem 2.6, N is semiregular, and hence ∣ N ∣ divides ∣ V Γ ∣ = 2 p q m . It follows that N is soluble and N ≅ Z 2 , Z p or Z q s with s ≤ n . For the last case, we are done. For the first case, by Theorem 2.6, Γ / N is connected arc-transitive of odd order p q n and odd valency 7, a contradiction. For the second case, again by Theorem 2.6, Γ / N is G / N -arc-transitive of order 2 q m and valency 7. Clearly, Γ / N ≠ K 7 , 7 . It then follows from Lemma 2.7 that either Γ / N = HS ( 50 ) , or G / N has a minimal normal subgroup M / N ≅ Z q k for some positive integer k . For the former case, ( p , q ) = ( 3 , 5 ) , and Γ is a Z 3 -cover of HS ( 50 ) . For the latter case, M = Z p . Z q k , and as p < q , the Sylow q -subgroup Z q k of M is characteristic in M , and hence normal in G .□

Now, we are ready to prove Theorems 1.1 and Corollary 1.2.

Proof of Theorems 1.1

Suppose Γ is G -arc-transitive with G ≤ Aut Γ . Let M be a maximal normal q -subgroup of G . Clearly, M has at least 2 p ≥ 6 orbits on V Γ . Then by Theorem 1.1, Γ is a normal cover of Γ / M , and Γ / M is a connected G / M -arc-transitive graph of order 2 p q m with 0 ≤ m ≤ n − 1 .

If m = 0 , then Γ / M is of order 2 p , by Lemma 2.3(1), Γ / M = K 7 , 7 or CD ( 2 p , 7 ) with 7 ∣ p − 1 .

If m = 1 , then Γ / M is of order 2 p q , by Lemma 2.3(2), Γ / M = K 8 , C 30 , C 78 1 , C 78 2 , C 310 , or CD ( 2 p q , 7 ) with 7 ∣ q − 1 . For the last case, as 2 p q > 31 , by Example 2.1, Γ / M = CD ( 2 p q , 7 ) is arc-regular, so G / M = A u t ( Γ / M ) ≅ D 2 p q : Z 7 . Now, G has a normal subgroup H = M . Z q , which has 2 p orbits on V Γ , and Γ is a normal cover of Γ / H = CD ( 2 p , 7 ) by Theorem 1.1.

Now assume m ≥ 2 . By Lemma 5.1, either Γ / M is a Z 3 -cover of HS ( 50 ) , or G has a normal elementary abelian q -subgroup, say X / M . For the former case, Γ is a normal M . Z 3 -cover of HS ( 50 ) . For the latter case, X is a normal q -subgroup of G , and by the maximality of M , X has at most two orbits on V Γ . It follows that p q n ∣ ∣ X ∣ , a contradiction.

Finally, one easily verifies that the graphs in Table 1 are basic graphs. This completes the proof of Theorems 1.1.□

Proof of Corollary 1.2

Let Γ be a connected 2-arc-transitive 7-valent graphs of order 2 p q n with q > p primes.

If n = 1 , note that CD ( 2 p q , 7 ) is not 2-arc-transitive, Corollary 1.2 is true by Lemma 2.3(2).

Assume n ≥ 2 . If p ≠ 7 , by Theorems 1.1 and 2.6, Γ is a normal cover of C 30 , HS ( 50 ) or C 310 , hence ( p , q ) = ( 3 , 5 ) or ( 5 , 31 ) . Now Corollary 1.2 easily follows.□

Acknowledgements

The authors thank the referees for their helpful comments.

  1. Funding information: This work was partially supported by the National Natural Science Foundation of China (11961076 and 11461007).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-09-13
Revised: 2021-06-13
Accepted: 2021-06-13
Published Online: 2021-08-06

© 2021 Jiangmin Pan et al., 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. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0062
Keywords for this article
symmetric graph; simple group; normal quotient graph
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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