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 5 publications

(12 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

Self Cite

View full text Add to dashboard Cite

show abstract

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

Vertex-primitive digraphs with large fixity

Barbieri,

Potočnik

2024

Annali di Matematica

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 5 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

Contact Info

Product

Resources

About