Vertex-transitive Automorphism Group Research Articles

Vertex-transitive graphs with local action the symmetric group on ordered pairs

Abstract We consider a finite, connected and simple graph Γ that admits a vertex-transitive group of automorphisms 𝐺. Under the assumption that, for all x ∈ V ⁢ ( Γ ) x\in V(\Gamma) , the local action G x Γ ⁢ ( x ) G_{x}^{\Gamma(x)} is the action of Sym ⁡ ( n ) \operatorname{Sym}(n) on ordered pairs, we show that the group G x [ 3 ] G_{x}^{[3]} , the point-wise stabiliser of a ball of radius three around 𝑥, is trivial.

Formula omitted]-regular families of graph automorphisms

An r-regular family F of permutations on a set V contains, for each pair of vertices u,v∈V, exactly r permutations φ mapping u to v. Earlier, 1-regular families of graph automorphisms were used by Gauyacq to define the quasi-Cayley graphs, a class of vertex-transitive graphs that properly contains the class of Cayley graphs, sharing many of their characteristics, and is properly contained in the class of vertex-transitive graphs. We introduce r-regular families to measure how far a vertex-transitive graph is from being quasi-Cayley. As any automorphism group of a graph Γ=(V,E) acting transitively on V with vertex-stabilizers of order r forms an r-regular family on V, every vertex-transitive graph admits an r-regular family of automorphisms for some r≥1. In general, the smallest r for which such a family exists (which we call the quasi-Cayley deficiency of the graph) might be smaller than the order of the vertex-stabilizer of a smallest vertex-transitive automorphism group of the graph (which we call the Cayley deficiency). We investigate the relations between these two parameters for the class of merged Johnson graphs. We prove the existence of Johnson graphs with arbitrarily large quasi-Cayley deficiency, as well as Johnson graphs for which the difference between their Cayley deficiency and their quasi-Cayley deficiency is arbitrarily large.

Non-commutative association schemes of rank 6 with affine subschemes

When association schemes are viewed as a generalization of groups, it becomes natural to seek non-commutative examples. As with groups, non-commutative association schemes must have at least six elements, but unlike in group theory, there are numerous examples with exactly six elements. One method to try to classify such schemes is to attempt to construct extensions of schemes of rank 3, starting with those schemes of rank 3 which correspond to self-complementary strongly regular graphs with a vertex-transitive automorphism group. Recent work of Klin, Kriger, and Woldar provides new constructions for such graphs. In this paper, we investigate the possibility of constructing new non-commutative schemes with six elements from these graphs.

Big subsets with small boundaries in a graph with a vertex-transitive group of automorphisms

The theory of ends of finitely generated groups and connected locally finite graphs with vertex- transitive groups of automorphisms can be regarded as a theory of Boolean algebras of subsets of or vertex set of with finite boundaries (in the locally finite Cayley graph of or in respectively), considered modulo finite subsets. We develop a more general theory where infinite subsets with finite boundaries are replaced by certain `big' subsets with `small' boundaries.

Big subsets with small boundaries in a graph with a vertex-transitive group of automorphisms

Теорию концов конечно порожденных групп $G$ и связных локально конечных графов $\Gamma$ с вершинно-транзитивными группами автоморфизмов можно интерпретировать как теорию булевых алгебр подмножеств группы $G$ или множества вершин графа $\Gamma$ с конечными границами (в локально конечном графе Кэли группы $G$ или в графе $\Gamma$ соответственно), рассматриваемых с точностью до конечных подмножеств. В настоящей статье развивается более общая теория, в которой бесконечные подмножества с конечными границами заменяются определенными "большими" подмножествами с "малыми" границами. Библиография: 7 наименований.

Semiregular Automorphisms of Cubic Vertex-Transitive Graphs and the Abelian Normal Quotient Method

We characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.

Some remarks on symmetrical extensions of graphs

It is proved that results from a previous paper of the author on symmetrical 2-extensions of graphs can be extended to symmetrical p-extensions of graphs for any prime p. In particular, it is proved that, for any prime p, there are only finitely many symmetrical p-extensions of a locally finite graph with an abelian subgroup of finite index in its automorphism group. Some refinements of these results are also obtained. In addition, we consider the question on the possibility to represent symmetrical extensions of a d-dimensional grid (and similar graphs) in the d-dimensional affine Euclidean space in such a way that a corresponding vertex-transitive group of automorphisms of the extension is induced by some crystallographic group of motions of the space.

Equivelar maps on the torus

We give a classification of all equivelar polyhedral maps on the torus. In particular, we classify all triangulations and quadrangulations of the torus admitting a vertex transitive automorphism group. These are precisely the ones which are quotients of the regular tessellations {3,6}, {6,3} or {4,4} by a pure translation group. An explicit formula for the number of combinatorial types of equivelar maps (polyhedral and non-polyhedral) with n vertices is obtained in terms of arithmetic functions in elementary number theory, such as the number of integer divisors of n . The asymptotic behaviour for n → ∞ is also discussed, and an example is given for n such that the number of distinct equivelar triangulations of the torus with n vertices is larger than n itself. The numbers of regular and chiral maps are determined separately, as well as the ones for all other kinds of symmetry. Furthermore, arithmetic properties of the integers of type p 2 + p q + q 2 (or p 2 + q 2 , resp.) can be interpreted and visualized by the hierarchy of covering maps between regular and chiral equivelar maps or type {3,6} (or {4,4}, resp.).

Degree-regular triangulations of the double-torus

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinatorial 2-manifold of Euler characteristic –2 must contain 12 vertices. In 1982, McMullen et al . constructed a 12-vertex geometrically realized triangulation of the double-torus in ℝ 3 . As an abstract simplicial complex, this triangulation is a weakly regular combinatorial 2-manifold. In 1999, Lutz showed that there are exactly three weakly regular orientable combinatorial 2-manifolds of Euler characteristic –2. In this article, we classify all the orientable degree-regular combinatorial 2-manifolds of Euler characteristic –2. There are exactly six such combinatorial 2-manifolds. This classifies all the orientable equivelar polyhedral maps of Euler characteristic –2.

Invariant subspaces, duality, and covers of the Petersen graph

A general method for finding elementary abelian regular covering projections of finite connected graphs is applied to the Petersen graph. As a result, a complete list of pairwise non-isomorphic elementary abelian covers admitting a lift of a vertex-transitive group of automorphisms is given. The resulting graphs are explicitly described in terms of voltage assignments.

On sharply vertex transitive 2-factorizations of the complete graph

We introduce the concept of a 2-starter in a group G of odd order. We prove that any 2-factorization of the complete graph admitting G as a sharply vertex transitive automorphism group is equivalent to a suitable 2-starter in G. Some classes of 2-starters are studied, with special attention given to those leading to solutions of some Oberwolfach or Hamilton–Waterloo problems.

One-point suspensions and wreath products of polytopes and spheres

It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way.

Nilpotent 1-factorizations of the complete graph

For which groups G of even order 2n does a 1-factorization of the complete graph K2n exist with the property of admitting G as a sharply vertex-transitive automorphism group? The complete answer is still unknown. Using the definition of a starter in G introduced in 4, we give a positive answer for new classes of groups; for example, the nilpotent groups with either an abelian Sylow 2-subgroup or a non-abelian Sylow 2-subgroup which possesses a cyclic subgroup of index 2. Further considerations are given in case the automorphism group G fixes a 1-factor. © 2005 Wiley Periodicals, Inc. J Combin Designs

Edge-colourings of cubic graphs admitting a solvable vertex-transitive group of automorphisms

It is proved that every connected cubic simple graph admitting a vertex-transitive action of a solvable group is either 3-edge-colourable, or isomorphic to the Petersen graph.

Constructing Homogeneous Factorisations of Complete Graphs and Digraphs

A homogeneous factorisation of a complete graph K n is a partition of the edge set that is invariant under a subgroup G of S n such that G is transitive on the parts of the partition and induces a vertex-transitive automorphism group on the graph corresponding to each part. A product construction is given for such factorisations.

Small vertex-transitive directed strongly regular graphs

We consider directed strongly regular graphs defined in 1988 by Duval. All such graphs with n vertices, n⩽20, having a vertex-transitive automorphism group, are determined with the aid of a computer. As a consequence, we prove the existence of directed strongly regular graphs for three feasible parameter sets listed by Duval. For one parameter set a computer-free proof of the nonexistence is presented. This, together with a recent result by J ørgensen, gives a complete answer on Duval's question about the existence of directed strongly regular graphs with n⩽20. The paper includes catalogues of all generated graphs and certain theoretical generalizations based on some known and new graphs.

A CENSUS OF TIGHT TRIANGULATIONS

A triangulation of a manifold (or pseudomanifold) is called a tight triangulation if any simplexwise linear embedding into any Euclidean space is tight. Tightness of an embedding means that the inclusion of any sublevel selected by a linear functional is injective in homology and, therefore, topologically essential. Tightness is a generalization of convexity, and the tightness of a triangulation is a fairly restrictive property. We give a review on all known examples of tight triangulations and formulate a (computer-aided) enumeration theorem for the case of at most 15 vertices and the presence of a vertex-transitive automorphism group. Altogether, six new examples of tight triangulations are presented, a vertex-transitive triangulation of the simply connected homogeneous 5-manifold SU(3)/SO(3) with vertex-transitive action, two non-symmetric 12-vertex triangulations of S3 × S2, and two non-symmetric triangulations of S3 × S3 on 13 vertices.

Graphs with Projective Linear Stabilizers

We investigate the structure of the free amalgamated productP1*P1∩P2P2in whichP1andP2are isomorphic projective linear groups andP1∩P2is a one-point stabilizer in the natural action ofPion the points of a projective space of dimensionn≥ 2. We apply the results to graphs admitting a vertex-transitive automorphism group such that the subgroup stabilizing a vertex is a projective linear group. In particular, graphs of girth 4 with projective linear stabilizers are characterized.

Vertex-Transitive Non-Cayley Graphs with Arbitrarily Large Vertex-Stabilizer

A construction is given for an infinite family {Γn} of finite vertex-transitive non-Cayley graphs of fixed valency with the property that the order of the vertex-stabilizer in the smallest vertex-transitive group of automorphisms of Γn is a strictly increasing function ofn . For each n the graph is 4-valent and arc-transitive, with automorphism group a symmetric group of large prime degree \(p < 2^{2^n + 2}\). The construction uses Sierpinski's gasket to produce generating permutations for the vertex-stabilizer (a large 2-group).

A note on bounded automorphisms of infinite graphs

LetX be a connected locally finite graph with vertex-transitive automorphism group. IfX has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup ofAUT(X) and the action ofAUT(X) onV(X) is imprimitive ifX is not finite. IfX has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.