On the Number of Fixed Points of Automorphisms of Vertex-Transitive Graphs

Potočnik, Primož; Spiga, Pablo

doi:10.1007/s00493-020-4509-y

Cited by 6 publications

(12 citation statements)

References 40 publications

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“…The main results of this paper and the results in [PS21b] show that, besides small exceptions or well-understood families of graphs, non-identity automorphisms of 3-valent or 4-valent vertextransitive graphs cannot fix many vertices or edges. Where "too many" in this context has to be considered as a linear function on the number of vertices (and, even then, with a small caveat for 4-valent graphs, because of the assumption of edge-transitivity).…”

Section: Introductionmentioning

confidence: 74%

“…Therefore fpr(V Γ, g) > 1/3. Now, the hypothesis of Lemma 2.3 in [PS21b] are satisfied. Therefore, [PS21b, Lemma 2.3] implies that Γ is a Praeger-Xu graph, which is our final contradiction.…”

Section: Lemma 32 Let γ Be a Finite Connected Graph And Letmentioning

confidence: 92%

“…Potočnik and Spiga have proved in [PS21b] that, if Γ is a finite connected 3-valent vertextransitive graph, or a 4-valent vertex-and edge-transitive graph then, unless Γ belongs to a well-known family of graphs, every non-identity automorphism of Γ fixes at most 1/3 of the vertices. In the same work, they have proposed a similar investigation with respect to the edges of the graph, see [PS21b,Problem 1.7]. In this paper we solve this problem.…”

Section: Introductionmentioning

confidence: 99%

“…Investigations on the number of fixed points of graph automorphisms do have interesting applications. For instance, very recently Potočnik, Toledo and Verret [PTV] pivoting on the results in [PS21b] have proved remarkable results on the cycle structure of general automorphisms of 3-valent vertex-transitive and 4-valent arc-transitive graphs.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency

Barbieri¹,

Grazian²,

Spiga³

2022

Preprint

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Section: Introductionmentioning

confidence: 74%

Section: Lemma 32 Let γ Be a Finite Connected Graph And Letmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency

Barbieri¹,

Grazian²,

Spiga³

2022

Preprint

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“…This lemma, together with a recent result of Pablo Spiga and the first-named author of this paper [26], which bounds the number of vertices that can be fixed by a non-trivial automorphism in a cubic-vertex transitive graph, yields the following. Corollary 5.2.…”

mentioning

confidence: 81%

On orders of automorphisms of vertex-transitive graphs

Potocnik,

Toledo,

Verret

2021

Preprint

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In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d, d ≤ 4, is at most c d n where c 3 = 1 and c 4 = 9. Whether such a constant c d exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, Γ ∼ = K 3,3 , has a regular orbit, that is, an orbit of g of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C.

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Vertex-primitive digraphs with large fixity

Barbieri,

Potočnik

2024

Annali di Matematica

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The relative fixity of a digraph $$\Gamma $$ Γ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $$\Gamma $$ Γ and the number of vertices of $$\Gamma $$ Γ . We characterize the vertex-primitive digraphs whose relative fixity is at least $$\frac{1}{3}$$ 1 3 , and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.

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On the Number of Fixed Points of Automorphisms of Vertex-Transitive Graphs

Cited by 6 publications

References 40 publications

On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency

On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency

On orders of automorphisms of vertex-transitive graphs

Vertex-primitive digraphs with large fixity

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