The nonexistence of 8-transitive graphs

Weiss, Richard M.

doi:10.1007/bf02579337

Cited by 154 publications

(110 citation statements)

References 14 publications

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“…Tutte's Theorem was generalized by Weiss [24] who proved that there exist no finite stransitive graphs of valency at least 3 for s = 6 and s ≥ 8. Since then, characterizing s-arc transitive graphs has received considerable attention in the literature (see, for example, [11,14,19,24]). …”

Section: Introductionmentioning

confidence: 99%

Finite 𝑠-arc transitive Cayley graphs and flag-transitive projective planes

2004

Proc. Amer. Math. Soc.

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Abstract. In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k = 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs with s ≥ 3 are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

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Section: Introductionmentioning

confidence: 99%

Finite 𝑠-arc transitive Cayley graphs and flag-transitive projective planes

2004

Proc. Amer. Math. Soc.

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“…This situation is quite different from that of finite undirected regular graphs of degree greater than 2, which by a theorem of Richard Weiss [7] can be at most 7-arc-transitive.…”

Section: Introductionmentioning

confidence: 76%

“…Also a~lx t a = x i+ i for 1 < / < s whereas none of the conjugates of the non-trivial powers terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700038477 [7] Constructions for arc-transitive digraphs 67 of x s+1 by the permutation a lie in H (because they all move the point (s + l)k + 1), therefore H n a' 1 Ha is the subgroup generated by all the Xj other than x u implying that this digraph A = A(G, H, a) is ^-regular. The intersection of all the conjugates of H in G is trivial, so G acts faithfully on A.…”

Section: Examplementioning

confidence: 99%

Constructions for arc-transitive digraphs

1995

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A number of constructions are given for arc-transitive digraphs, based on modifications of permutation representations of finite groups. In particular, it is shown that for every positive integer .s and for any transitive permutation group P of degree k, there are infinitely many examples of a finite k-regular digraph with a group of automorphisms acting transitively on s-arcs (but not on (s + l)-arcs), such that the stabilizer of a vertex induces the action of P on the out-neighbour set.1991 Mathematics subject classification (Amer. Math. Soc): 05C25, 20B25.

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“…These three classes of graphs are important because of the following result (see [10] and [33,35], also refer to [13, Theorem 2.1.5]).…”

Section: Amalgams Of S-transitive Graphs For S ∈ {4 5 7}mentioning

confidence: 99%

The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$

2001

Trans. Amer. Math. Soc.

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Abstract. A complete classification is given for finite vertex-primitive and vertex-biprimitive s-transitive graphs for s ≥ 4. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group M, and an infinite family of graphs of valency 5 admitting projective symplectic groups PSp(4, p) with p prime and p ≡ ±1 (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.

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The nonexistence of 8-transitive graphs

Cited by 154 publications

References 14 publications

Finite 𝑠-arc transitive Cayley graphs and flag-transitive projective planes

Finite 𝑠-arc transitive Cayley graphs and flag-transitive projective planes

Constructions for arc-transitive digraphs

The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$

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