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Classifying cubic symmetric graphs of order 88p and 88p 2

  • Liangliang Zhai EMAIL logo
Published/Copyright: May 13, 2025
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Open Mathematics
From the journal Open Mathematics Volume 23 Issue 1

Abstract

For a simple graph Γ , Γ is said to be s -regular, provided that the automorphism group of Γ regularly acts on the set consisting of s -arcs of Γ . Given a positive integer n , the question on finding all s -regular graphs of order n and degree 3 has received considerable attention. An s -regular graph with degree 3 is so-called a cubic symmetric graph. Let p be a prime. We show that if Γ is a cubic symmetric graph of order 88 p , then p ∈ { 5 , 11 , 23 } ; if Γ is a cubic symmetric graph of order 88 p 2 , then p = 11 . Moreover, we classify all cubic symmetric graphs of order 88 p and 88 p 2 .

Keywords: cubic symmetric graph; simple group
MSC 2010: 05C25

1 Introduction

In the short note, it is assumed that every graph being discussed is finite and connected. For a given graph, say X , V ( X ) always denotes the vertex set of X , and E ( X ) denotes the edge set of X . In this graph X , the ordered ( s + 1 ) -tuple ( w 0 , w 1 , … , w s ) , where { w 0 , w 1 , … , w s } ⊆ V ( X ) that satisfies what w i − 1 is adjacent to w i for any 1 ≤ i ≤ s and w i − 1 ≠ w i + 1 for every 1 ≤ i < s is called an s-arc. For given graph X , the automorphism group of X is denoted by Aut ( X ) . If Aut ( X ) acting on the set consisting of all s -arcs of X is transitive, then X is said to be s-arc-transitive. Particularly, 0-arc-transitive is also said to be vertex-transitive. Also, a 1-arc-transitive graph is said to be arc-transitive, and sometimes, we also called it as a symmetric graph.

Assume that G is a permutation group. G that acts on the set Ω is called semiregular, provided that no non-trivial element of G can fix a point of Ω . Thus, from orbit-stabilizer theorem, it follows that if G is a semi-regular group, then the length of any orbit is ∣ G ∣ . In particular, if G is transitive and semi-regular, then G is said to be a regular group. Recall that X is graph. If a subgroup of Aut ( X ) acts regularly on the set consisting of all s -arcs of X , then this subgroup is called a s-regular subgroup. In particular, if we take the subgroup which is Aut ( X ) , then the given graph X is called a s-regular graph. Given a group G and a subset S of G , if S − 1 ≔ { s − 1 : s ∈ S } = S , then S is called a inverse-closed subset. Now, suppose that e is the identity of G , and S is a subset of G and is inverse-closed with e ∉ S . The Cayley graph, denoted by Cay ( G , S ) , is a simple graph with vertex set G where two distinct a , b are adjacent, provided that b a − 1 ∈ S .

Suppose that N is a subgroup of Aut ( X ) where X is a graph. The quotient graph with N , denoted by X N , is a graph whose vertex set is the set of all orbits of N , and two distinct orbits A and B are adjacent in X N if and only if in graph X , there is an edge whose one vertex is in A and another vertex in B . Recall that in a graph, N ( v ) is the set consisting of all vertices adjacent to this vertex v . For two graphs X ˜ and X , X ˜ is said to be a covering of X (with a mapping, say ρ , from X ˜ to X , provided that this mapping ρ is surjective from V ( X ˜ ) to V ( X ) satisfying that ρ ∣ N ( v ˜ ) from N ( v ˜ ) to N ( v ) is bijective for each two vertices v ∈ V ( X ) and v ˜ ∈ ρ − 1 ( v ) . Assume that ρ : X ˜ → X is a covering. If this group Aut ( X ˜ ) has a semi-regular subgroup, say N , so that X ≅ X ˜ N , then ρ is said to be a regular covering.

The next result from [1, Theorem 9] will be used frequently in this article.

Proposition 1.1

Suppose that X is cubic and symmetric. For s ≥ 1 , suppose that G is an s-regular subgroup of Aut ( X ) . If G has a normal subgroup, say N, which has more than two orbits acting on V ( X ) , then N is semi-regular and this quotient group G ⁄ N is s-regular subgroup of Aut ( X N ) . In particular, in this case, X is a regular covering of X N .

In [2,3], Tutte proved that any cubic symmetric graph is also an s -regular graph, where 1 ≤ s ≤ 5 . All cubic symmetric graphs with order 2 p were classified by Cheng and Oxley [4]. Since then, for a given number n , the classifying cubic symmetric graphs of order n became an interesting research topic, see [5–9].

We use Z n to denote a cyclic group of order n . Let G 1 = Z 2 2 ⋉ ( Z 11 3 × Z 2 ) and G 2 = Z 2 ⋉ ( ( Z 11 ⋉ Z 11 2 ) × Z 2 2 ) . Write

(1) G 1 ≔ Cay ( G 1 , { ( 1 , 0 , − 1 , 0 , 0 , 1 ) , ( 0 , 1 , 0 , − 1 , 0 , 1 ) , ( 1 , 1 , 0 , 0 , − 1 , 1 ) } ) , G 2 ≔ Cay ( G 2 , { ( 1 , 0 , 0 , 0 , 0 , 0 ) , ( 1 , 0 , 1 , 0 , 1 , 0 ) , ( 1 , 1 , 0 , 0 , 0 , 1 ) } ) .

In 2014, Feng et al. [5] proved that both G 1 and G 2 are cubic 2-regular graphs of order 10648.

In 2011, with the help of a computer, Conder obtained all cubic symmetric graphs of order at most 10000 and uploaded the results on the website [10]. With the help of [10], this article classifies the cubic symmetric graphs with order 88 p and 88 p 2 , where p is a prime. The main results of our article are the following:

Theorem 1.2

Suppose that p is a prime and X is a cubic symmetric graph with order 88 p . Then, X is isomorphic to one of these graphs in Table 1 from [10].

Table 1

All cubic symmetric graphs of order 88 p

Graph Order ∣ Aut ( X ) ∣ Girth Diameter s -Regular
C440.1 88 ⋅ 5 2640 10 12 2
C440.2 88 ⋅ 5 2640 10 11 2
C440.3 88 ⋅ 5 5280 12 10 3
C968.1 88 ⋅ 11 5808 6 29 2
C2024.1 88 ⋅ 23 12144 14 13 2
C2024.2 88 ⋅ 23 12144 11 14 2
C2024.3 88 ⋅ 23 12144 14 13 2
C2024.4 88 ⋅ 23 12144 12 14 2
C2024.5 88 ⋅ 23 12144 8 15 2
C2024.6 88 ⋅ 23 12144 12 14 2
C2024.7 88 ⋅ 23 12144 11 14 2
C2024.8 88 ⋅ 23 12144 8 14 2
C2024.9 88 ⋅ 23 12144 11 12 2
C2024.10 88 ⋅ 23 12144 11 15 2
C2024.11 88 ⋅ 23 12144 12 13 2
C2024.12 88 ⋅ 23 12144 12 12 2
C2024.13 88 ⋅ 23 24288 16 16 3

Theorem 1.3

Assume that p is a prime and X is a cubic symmetric graph with order 88 p 2 . Then, X is isomorphic to either G 1 or G 2 in (1).

We will prove Theorems 1.2 and 1.3 in Sections 2 and 3, respectively.

2 Proof of Theorem 1.2

In the following, we always assume that graph X is cubic and symmetric. Recall that Aut ( X ) acting on this set consisting of all s -arcs of X is transitive, where 1 ≤ s ≤ 5 . Note that the size of the set consisting of all s -arcs of X is equal to 2 s − 1 ⋅ 3 ⋅ ∣ V ( Γ ) ∣ , we have

(2) ∣ Aut ( X ) ∣ = 2 s − 1 ⋅ 3 ⋅ ∣ V ( X ) ∣ .

In the following, we first prove Theorem 1.2.

Proof of Theorem 1.2

Note that p is a prime. If p < 127 , then X has order at most 10000, and so by Conder [10], X is isomorphic to one of these graphs in Table 1, as desired.

Suppose next that p ≥ 127 . In order to show Theorem 1.2, it suffices to prove that such a graph X does not exist. Suppose, for a contradiction, that X is a cubic symmetric graph with order 88 p . Note that it follows from (2) that we have

∣ Aut ( X ) ∣ = 2 s + 2 ⋅ 3 ⋅ 11 ⋅ p , 1 ≤ s ≤ 5 .

Let A = Aut ( X ) and N be a minimal normal subgroup of A . We shall divide our proof into four steps.

Step 1. A has no normal p -subgroups.

Suppose that P is a normal p -subgroup of A . Then, ∣ P ∣ = p , which implies that P has more than two orbits on V ( X ) . By Proposition 1.1, we have that X P is a cubic symmetric graph of order 88, this is in contradiction to [10].

Step 2. A has no normal 2-subgroups.

Suppose that H is a normal 2-subgroup of A . By Proposition 1.1, X N is a cubic symmetric graph, and H on V ( X ) is semiregular. As a consequence, ∣ H ∣ is a divisor of ∣ X ∣ , and therefore, we have ∣ H ∣ = 2 , 4, or 8. If ∣ H ∣ = 2 , then X H is of order 44 p , which is impossible by [8, Theorem 2.4]. If ∣ H ∣ = 4 , then X H is of order 22 p , which is impossible by [9, Theorem 3.3]. If ∣ H ∣ = 8 , then ∣ V ( X H ) ∣ is odd, which is impossible.

Step 3. N is solvable.

Suppose that N is non-solvable. Then, N is a direct product of several copies of a non-abelian simple group T . Observe that any prime factor of ∣ T ∣ belongs to { 2 , 3 , 11 , p } . If p > 2 7 ⋅ 3 ⋅ 11 , then A has a normal p -subgroup, it obtains a contradiction by Step 1. Therefore, ∣ T ∣ ≤ ∣ A ∣ < 1 0 25 . By [11, pp. 239–242], the simple group T does not exist, it obtains a contradiction.

Step 4. Final contradiction.

By Step 3, we see that N must be an elementary abelian r -group where r is prime. It follows from Proposition 1.1 that N acting on V ( X ) must be semi-regular, which implies that ∣ N ∣ is a divisor of ∣ X ∣ . Therefore, N ≅ Z 11 by Steps 1 and 2. Let J ⁄ N be a minimal normal subgroup of A ⁄ N . Since ∣ A ⁄ N ∣ = 2 s + 2 ⋅ 3 ⋅ p , it is similar to Step 3, one has that J ⁄ N is elementary abelian. Since J ⁄ N is semi-regular on V ( X N ) , 8 p is divided by ∣ J ⁄ N ∣ . It follows that J ⁄ N ≅ Z p or J ⁄ N is a 2-group. If J ⁄ N ≅ Z p , then ∣ J ∣ = 11 p and so J has a normal Sylow p -subgroup P . Since P is characteristic in J and J is normalized by A , P is normal in A , contrary to Step 1. So, J ⁄ N is a 2-group. If ∣ J ⁄ N ∣ = 8 , then X J is a cubic graph on p vertices, which is impossible. So, ∣ J ⁄ N ∣ = 2 or 4. If ∣ J ⁄ N ∣ = 2 , then X J is a cubic symmetric graph of order 4 p , contradicting [6, Theorem 6.2]. Now, suppose ∣ J ⁄ N ∣ = 4 . Then, X J is a cubic symmetric graph of order 2 p . Let K ⁄ J be a minimal normal subgroup of A ⁄ J . It is similar to Step 3, K ⁄ J is elementary abelian. Then, ∣ K ⁄ J ∣ = p and ∣ K ∣ = 44 p . Also, by the proof of Case 1, we have a contradiction.

By the above discussion, we end the proof of Theorem 1.2.□

3 Proof of Theorem 1.3

Given a group G , let H be a subgroup of G . The center of G is denoted by Z ( G ) . We use G ′ to denote the derived subgroup of G . The symbol [ G : H ] denotes the index of subgroup H in G , which is equal to the number of all right (or left) cosets of H of G . Two integers a and b are said to be coprime if the only positive factor that divides both of them is 1, and we denote it by ( a , b ) = 1 .

Lemma 3.1

Suppose that G is a group. Let H be a subgroup of G with ( [ G : H ] , [ H : H ′ ] ) = 1 . Then

H ∩ G ′ ∩ Z ( G ) ⊆ H ′ .

Proof

Take x ∈ H ∩ G ′ ∩ Z ( G ) . According to the transfer from G to H ⁄ H ′ (cf. [12, Chapter 10]), one has x [ G : H ] ∈ H ′ . Further, since x H ′ ∈ H ⁄ H ′ , it follows that ( x H ′ ) [ H : H ′ ] = H ′ , which also implies x [ H : H ′ ] ∈ H ′ . Now in view of

( [ G : H ] , [ H : H ′ ] ) = 1 ,

we have x ∈ H ′ , as desired.□

In group G , the largest normal p -subgroup is denoted by O p ( G ) .

Lemma 3.2

Suppose that p is a prime at least 13. There is no cubic symmetric graph with order 44 p 2 .

Proof

Suppose, to the contrary, that there exists a cubic symmetric graph, say X , which has order 44 p 2 . We write A = Aut ( X ) . Assume that P is a Sylow p -subgroup of A . Then

∣ A ∣ = 2 s + 1 ⋅ 3 ⋅ 11 ⋅ p 2 , 1 ≤ s ≤ 5 .

In view of [10], one has that P is not normal in A .

Since the number of orbits of O 2 ( A ) acting on V ( X ) is greater than 2, 44 p 2 is divisible by ∣ O 2 ( A ) ∣ . It follows that ∣ O 2 ( A ) ∣ = 1 , 2, or 4. If ∣ O 2 ( A ) ∣ = 2 , one has ∣ X O 2 ( A ) ∣ = 22 p 2 , which is a contradiction by [9, Theorem 3.4]. If ∣ O 2 ( A ) ∣ = 4 , then X O 2 ( A ) is a cubic graph with odd order, which is a contradiction. Therefore, O 2 ( A ) = 1 . Similarly, by [6, Theorem 6.2], we obtain O 11 ( A ) = 1 .

Suppose, now, that T is a minimal normal subgroup of A . If T is nonsolvable, then, by [11, pp. 239–242], one has T ≅ P S L ( 2 , 23 ) or P S L ( 2 , 23 ) . Since T on V ( X ) has more than two orbits, ∣ X ∣ is divisible by ∣ T ∣ , which is impossible. As a result, T is an elementary abelian group, which implies that T ≅ Z p .

Now note that X T is cubic and symmetric, which has order 44 p . In view of [8, Theorem 2.4], one has that X T is 2- or 3-regular and p = 23 . Thus, A is at most 3-regular. As a consequence, ∣ A ∣ is a divisor of 2 4 ⋅ 3 ⋅ 11 ⋅ p 2 . Hence, A has p + 1 Sylow p -subgroups.

Suppose that N is the normalizer of P in A . Now, let A act on the set of all right cosets of N of A , by right multiplication. Then, A ⁄ N A can be imbedded in the symmetric group on p + 1 letters, where N A is the largest normal subgroup of A contained in N . It means that ∣ A ⁄ N A ∣ is a divisor of ( p + 1 ) ! . Since p 2 ∣ ∣ A ∣ , one has p ∣ ∣ N A ∣ . If p 2 ∣ ∣ N A ∣ , the fact that ∣ A ⁄ N ∣ = 24 implies that ∣ N A ∣ ≤ ∣ N ∣ ≤ 2 ⋅ 11 ⋅ p 2 . Therefore, N A has a characteristic Sylow p -subgroup of order p 2 . Since N A is normalized by A , one has that P is normal in A , which is a contradiction. Thus, ∣ N A ∣ is not divisible by p 2 . This forces that the number of orbits under the action N A on V ( X ) is greater than 2. Proposition 1.1 implies that ∣ N A ∣ is a divisor of 22 p .

Let K be a Sylow p -subgroup in N A . It follows that K is normal in A . Also, in A , we say that C is the centralizer of K . By, N ⁄ C theorem (see, for example, [12, Theorem 1.6.13]), we know that N A ( K ) ⁄ C = A ⁄ C is isomorphic to a subgroup of Aut ( K ) . Since K ≅ Z p , we have Aut ( K ) ≅ Z p − 1 , and so ∣ A ⁄ C ∣ is a divisor of p − 1 . So, p 2 ∣ ∣ C ∣ . It is straightforward that C ′ ∩ K = K or 1. If C ′ ∩ K = K , then K ≤ C ′ . Since K ≤ Z ( C ) , p ∣ ∣ C ′ ∩ Z ( C ) ∣ . Let P 1 be a Sylow p -subgroup of C . Then, p ∣ ∣ P 1 ∩ C ′ ∩ Z ( C ) ∣ . However, by Lemma 3.1, P 1 ∩ C ′ ∩ Z ( C ) = 1 , which is a contradiction. Thus, C ′ ∩ K = 1 and so C ′ is not divisible by p 2 . It follows that C ′ is semiregular. As a result, ∣ C ′ ∣ ∣ 44 p . Now, suppose that H ⁄ C ′ is a Sylow p -subgroup of C ⁄ C ′ . By p 2 ∣ ∣ C ∣ , one has p 2 ∣ ∣ H ∣ . It follows that ∣ H ∣ ∣ 44 p 2 . Thus, H has a normal Sylow p -subgroup. In view of the commutativity of C ⁄ C ′ , one has that P is normal in A , which is impossible.□

Finally, we prove Theorem 1.3.

Proof of Theorem 1.3

If p ≤ 7 , in view of [10], there is no cubic symmetric graph with order 88 p 2 . If p = 11 , in view of [5, Theorem 6.1], X is isomorphic to G 1 or G 2 .

Now, suppose that p ≥ 13 . Write A = Aut ( X ) . Let P be a Sylow p -subgroup of A . Then, ∣ A ∣ = 2 s + 2 ⋅ 3 ⋅ 11 ⋅ p 2 , where 1 ≤ s ≤ 5 . Note that there is no cubic symmetric graph with order 88. It follows that P is non-normal in A .

Suppose O 2 ( A ) ≠ 1 . Then, ∣ O 2 ( A ) ∣ = 2 or 4. Thus, X O 2 ( A ) is a cubic symmetric graph of order 44 p 2 or 22 p 2 , contradicting Lemma 3.2 or [9, Theorem 3.4], respectively. Hence, O 2 ( A ) = 1 .

Suppose O 11 ( A ) ≅ Z 11 . Then, X O 11 ( A ) is cubic and symmetric, which has order 8 p 2 . In view of [7, Theorem 5.2], we have that X O 11 ( A ) is either cyclic or elementary abelian cover of the hypercube. It means that A ⁄ O 11 ( A ) has a normal Sylow p -subgroup, say M ⁄ O 11 ( A ) . Thus, ∣ M ∣ = 11 p 2 . Since M is normal in A , P is normal in A , which is a contradiction. As a result, O 11 ( A ) = 1 .

Now, let N be a minimal normal subgroup of A . If N is nonsolvable, in view of [11, p. 239], N ≅ PSL (2, 23) or PSL(2, 32). Now, by Proposition 1.1, we have that ∣ X ∣ must be divisible by ∣ N ∣ , which is impossible. As a result, N must be an elementary abelian group. As mentioned in the previous paragraphs, one has N ≅ Z p . Thus, X N is a cubic symmetric graph with order 88 p . In view of Theorem 1.2, one has p = 23 and X N is 2- or 3-regular. Thus, A is at most 3-regular. Therefore, ∣ A ∣ is a divisor of 2 5 ⋅ 3 ⋅ 11 ⋅ p 2 . This forces that A has p + 1 Sylow p -subgroups. Now, it is similar to the last two paragraphs of the proof of Lemma 3.2, we can also obtain a contradiction.

Based on the discussion, we complete the proof of Theorem 1.3.□

4 Conclusions

For a positive integer n , the question on classifying s -regular graphs of order n and degree 3 has received considerable attention. A s -regular graph with degree 3 is so-called a cubic symmetric graph. It was proved that every cubic symmetric graph is also a s -regular graph, where 1 ≤ s ≤ 5 . For some prime p and a graph Γ , if Γ is a cubic symmetric graph of order 88 p , this article showed that p ∈ { 5 , 11 , 23 } . Moreover, if Γ is a cubic symmetric graph of order 88 p 2 , this article showed that p = 11 . In fact, this article classified all cubic symmetric graphs of order 88 p and 88 p 2 for each prime p .

Acknowledgment

The author is deeply grateful to the reviewers for their valuable comments that improved the manuscript.

  1. Funding information: Author states no funding involved.

  2. Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and manuscript preparation.

  3. Conflict of interest: The author declares that there are no conflicts of interest regarding the publication of this article.

  4. Data availability statement: No datasets were generated or analyzed during the current study.

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Received: 2024-06-13
Revised: 2025-02-17
Accepted: 2025-04-10
Published Online: 2025-05-13

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  87. Characterization of derivations on strongly double triangle subspace lattice algebras by local actions
  88. Fixed point approaches to the stability of Jensen’s functional equation
  89. Geometry
  90. The regularity of solutions to the Lp Gauss image problem
  91. Solving the quartic by conics
  92. Group Theory
  93. On a question of permutation groups acting on the power set
  94. A characterization of the translational hull of a weakly type B semigroup with E-properties
  95. Harmonic Analysis
  96. Eigenfunctions on an infinite Schrödinger network
  97. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  98. Subharmonic functions and associated measures in ℝn
  99. Mathematical Logic, Model Theory and Foundation
  100. A topology related to implication and upsets on a bounded BCK-algebra
  101. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  102. Number Theory
  103. Fibonacci vector and matrix p-norms
  104. Recurrence for probabilistic extension of Dowling polynomials
  105. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  106. The number of rational points of some classes of algebraic varieties over finite fields
  107. Classification and irreducibility of a class of integer polynomials
  108. Decompositions of the extended Selberg class functions
  109. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  110. Fibonacci Cartan and Lucas Cartan numbers
  111. Recurrence relations satisfied by some arithmetic groups
  112. The hybrid power mean involving the Kloosterman sums and Dedekind sums
  113. Numerical Methods
  114. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  115. A kind of univariate improved Shepard-Euler operators
  116. Probability and Statistics
  117. Statistical inference and data analysis of the record-based transmuted Burr X model
  118. Multiple G-Stratonovich integral in G-expectation space
  119. p-variation and Chung's LIL of sub-bifractional Brownian motion and applications
  120. Real Analysis
  121. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  122. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  123. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  124. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  125. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  126. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  127. Topology
  128. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  129. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  130. On SI2-convergence in T0-spaces
  131. Generalized quandle polynomials and their applications to stuquandles, stuck links, and RNA folding
https://doi.org/10.1515/math-2025-0153
Keywords for this article
cubic symmetric graph; simple group
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  88. Fixed point approaches to the stability of Jensen’s functional equation
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  109. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
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  111. Recurrence relations satisfied by some arithmetic groups
  112. The hybrid power mean involving the Kloosterman sums and Dedekind sums
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  115. A kind of univariate improved Shepard-Euler operators
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  118. Multiple G-Stratonovich integral in G-expectation space
  119. p-variation and Chung's LIL of sub-bifractional Brownian motion and applications
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  124. The evaluation of a definite integral by the method of brackets illustrating its flexibility
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  129. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
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