Finite hexavalent edge-primitive graphs

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Abstract

Weiss (1973) determined all cubic edge-primitive graphs, and Guo, Feng and Li recently determined all tetravalent and pentavalent edge-primitive graphs (notice that their method is difficult to treat the bigger valency case because the edge stabilizers may be insoluble). In this paper, we study hexavalent edge-primitive graphs by using line graphs. The s-arc-transitivity of such graphs are determined, and the automorphism groups of such graphs besides K6,6 are proved to be almost simple. Two families of hexavalent edge-primitive graphs are also completely determined.

Introduction

For a finite and undirected graph Γ, the expression AutΓ denotes the group of all of its automorphisms. If there is an automorphism group GAutΓ that acts transitively or primitively on , the set of its edges (unordered pairs of adjacent vertices), then Γ is called edge-transitive or edge-primitive; if G is transitive on its vertex set or arc set (the set of its unordered set of adjacent vertices), then Γ is called G-vertex-transitive or G-arc-transitive. Edge-primitive regular graphs are all arc-transitive, see [7, Lemma 3.4]. Similarly, we say Γ is (G, s)-arc-transitive for a positive integer s if G is transitive on the set of its s-arcs, and Γ is (G, s)-transitive if G is transitive on the set of its s-arcs but not transitive on the set of its (s+1)-arcs. In particular, a graph that is (AutΓ,s)-transitive is simply called s-transitive.

The first work regarding edge primitive graphs was obtained by Weiss in 1973 [20] who determined all edge-primitive graphs of valency 3. The study of edge-primitive graphs was reinvigorated in 2010 by Giudici and Li [7] by establishing a general structure of such graphs, which has motivated a few classifications of such graphs, see [8] for valency 4 and [9] for valency 5 (note that the edge stabilizers are soluble for the two cases), and [16] for any prime valency having soluble edge stabilizers. Moreover, Li and Zhang [14] classifies edge-primitive 4-arc-transitive graphs (the edge-stabilizers are soluble for the graphs) by classifying finite primitive groups with soluble stabilizers. It seems challenging for classifying general edge-primitive graphs with valency bigger than 5 because the edge stabilizers are not necessarily soluble and the methods in [8], [9], [14], [16], [20] are generally not efficient.

This paper is aiming to study hexavalent edge-primitive graphs by using line graphs (it is the first time for line graphs being used to investigate edge-primitive graphs). Our main results are as following. Throughout this paper the notations are follow [1].

Theorem 1.1

Let Γ be a connected hexavalent edge-primitive graph. Then Γ is s-transitive with 2 ≤ s ≤ 4, and AutΓ is almost simple except Γ=K6,6.

Theorem 1.2

Let Γ be a connected hexavalent edge-primitive graph, and let α ∈ VΓ and e ∈ EΓ.

  • (1)

    If Γ is of odd order, then (s,AutΓ,AutΓα,AutΓe) is listed in Rows 1 and 2 of Table 1.

  • (2)

    If Γ is of even order and AutΓe is soluble, then (s,AutΓ,AutΓα,AutΓe) is listed in Rows 35 of Table 1.

We notice that the graph is the complete graph K7 in Row 1 of Table 1, and the graphs in other Rows can be specifically constructed by using coset graphs.

Section snippets

Preliminary results

We begin with an observation regarding the vertex stabilizers (also their Sylow 2-subgroups) and edge stabilizers of 2-arc-transitive hexavalent graphs, which is based on the list of the vertex stabilizers of arc-transitive hexavalent graphs obtained independently in [10, Theorem 3.1] and [15, Theorem 3.4].

Proposition 2.1

Let Γ be a connected hexavalent (G, s)-transitive graph with GAutΓ and s ≥ 2. Then s ≤ 4, and the triple (Gα, Ge, (Gα)2) lies in Table 2, where α ∈ VΓ, e ∈ EΓ and (Gα)2 denotes a Sylow

Line graphs and technical lemmas

The line graphs play an important role in the subsequent discussion. Given a graph Σ, its line graph is denoted by L(Σ), which is with the vertex set and two vertices e1, e2 are adjacent if and only if e1 and e2 are incident in Σ (namely they share a common vertex). If Σ is a regular graph, it is easy to see that L(Σ) is also regular with valency val(L(Σ))=2(val(Σ)1); also, an automorphism group G of Σ may naturally induce an automorphism group of L(Σ) via the way:{α,β}g={αg,βg}forall{α,β}E

Proof of theorem 1.1

For a transitive permutation group G on a set Ω, an orbital graph of G is with the vertex set Ω and the arc set (α, β)G where α, β ∈ Ω. Clearly, the group G can be viewed as an arc-transitive automorphism group of its orbital graphs.

Lemma 4.1

If Γ is a connected hexavalent G-edge-primitive graph with GAutΓ, then Γ is (G, s)-transitive with 2 ≤ s ≤ 4.

By assumption and [7, Lemma 3.4], Γ is G-arc-transitive, and using Proposition 2.1, we find Γ is s-transitive with s ≤ 4.

Suppose on the contrary that Γ is

Proof of theorem 1.2

Let Γ be a connected hexavalent G-edge-primitive graph of odd order. By Theorem 1.1, G is 2-arc-transitive on Γ and is almost simple, say with socle T. The following lemma further determines T.

Lemma 5.1

If Γ is of odd order, then Γ is (T, 2)-transitive, and TA7 or PSL(2,q) with q a prime power.

Proof

Since Γ is G-edge-primitive, T is transitive on , and in turn T is transitive on because Γ is of odd order, hence G=TGα for α ∈ . Since the outer automorphism groups of simple groups are soluble, so isGα/Tα=

Acknowledgments

The authors are grateful to the referees for their helpful comments.

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This work was partially supported by National Natural Science Foundation of China (11901512, 11961076). National Science Foundation of Yunnan Province (2019FD116).

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