Finite hexavalent edge-primitive graphs☆
Introduction
For a finite and undirected graph Γ, the expression denotes the group of all of its automorphisms. If there is an automorphism group that acts transitively or primitively on EΓ, 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 VΓ or arc set AΓ (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 -arcs. In particular, a graph that is -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 is almost simple except . Theorem 1.2 Let Γ be a connected hexavalent edge-primitive graph, and let α ∈ VΓ and e ∈ EΓ. If Γ is of odd order, then is listed in Rows 1 and 2 of Table 1. If Γ is of even order and is soluble, then is listed in Rows of Table 1.
We notice that the graph is the complete graph 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 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 which is with the vertex set EΣ 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 is also regular with valency ; also, an automorphism group G of Σ may naturally induce an automorphism group of via the way:
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 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 or with q a prime power. Proof Since Γ is G-edge-primitive, T is transitive on EΓ, and in turn T is transitive on VΓ because Γ is of odd order, hence for α ∈ VΓ. Since the outer automorphism groups of simple groups are soluble, so is
Acknowledgments
The authors are grateful to the referees for their helpful comments.
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