scholarly journals On Embeddings of Circulant Graphs

10.37236/4013 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Marston Conder ◽  
Ricardo Grande
Keyword(s):  
Cayley Graph ◽  
Cyclic Group ◽  
Circulant Graphs ◽  
Transitive Action ◽  
Planar Case ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs ◽  
Orientable Surfaces

A circulant of order $n$ is a Cayley graph for the cyclic group $\mathbb{Z}_n$, and as such, admits a transitive action of $\mathbb{Z}_n$ on its vertices. This paper concerns 2-cell embeddings of connected circulants on closed orientable surfaces. Embeddings on the sphere (the planar case) were classified by Heuberger (2003), and by a theorem of Thomassen (1991), there are only finitely many vertex-transitive graphs with minimum genus $g$, for any given integer $g \ge 3$. Here we completely determine all connected circulants with minimum genus 1 or 2; this corrects and extends an attempted classification of all toroidal circulants by Costa, Strapasson, Alves and Carlos (2010).

scholarly journals Cubic Non-Cayley Vertex-Transitive Bi-Cayley Graphs over a Regular $p$-Group

10.37236/3915 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Jin-Xin Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Equal Size ◽  
Group Of Automorphisms ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits of equal size. In this paper, we give a characterization of cubic non-Cayley vertex-transitive bi-Cayley graphs over a regular $p$-group, where $p>5$ is an odd prime. As an application, a classification of cubic non-Cayley vertex-transitive graphs of order $2p^3$ is given for each prime $p$.

scholarly journals Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

2014 ◽  
Vol 157 (1) ◽  
pp. 45-61
Author(s):  
PABLO SPIGA
Keyword(s):  
Automorphism Group ◽  
Positive Solution ◽  
Cayley Graph ◽  
Burnside Problem ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Restricted Burnside Problem ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractIn this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

When is the cayley graph of a semigroup isomorphic to the cayley graph of a group

2017 ◽  
Vol 67 (1) ◽  
Author(s):  
Shoufeng Wang
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Inverse Semigroups ◽  
Graphs Of Groups ◽  
Cayley Graphs Of Semigroups ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

AbstractIt is well known that Cayley graphs of groups are automatically vertex-transitive. A pioneer result of Kelarev and Praeger implies that Cayley graphs of semigroups can be regarded as a source of possibly new vertex-transitive graphs. In this note, we consider the following problem: Is every vertex-transitive Cayley graph of a semigroup isomorphic to a Cayley graph of a group? With the help of the results of Kelarev and Praeger, we show that the vertex-transitive, connected and undirected finite Cayley graphs of semigroups are isomorphic to Cayley graphs of groups, and all finite vertex-transitive Cayley graphs of inverse semigroups are isomorphic to Cayley graphs of groups. Furthermore, some related problems are proposed.

scholarly journals Classification of vertex-transitive graphs of order a prime cubed — I

2000 ◽  
Vol 224 (1-3) ◽  
pp. 99-106 ◽  
Author(s):  
Edward Dobson
Keyword(s):  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

scholarly journals Presentations for vertex-transitive graphs

2021 ◽  
Author(s):  
Agelos Georgakopoulos ◽  
Alex Wendland
Keyword(s):  
Cayley Graph ◽  
Group Presentation ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractWe generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex-transitive graph.

A Construction for Vertex-Transitive Graphs

1982 ◽  
Vol 34 (2) ◽  
pp. 307-318 ◽  
Author(s):  
Brian Alspach ◽  
T. D. Parsons
Keyword(s):  
Permutation Groups ◽  
General Strategy ◽  
Circulant Graphs ◽  
Cyclic Groups ◽  
Disjoint Cycles ◽  
The Family ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group Γ in the family all graphs G whose vertex–set is the orbit V of Γ and for which Γ ≦ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of cyclic groups 〈(0 1 … n – 1)〉 generated by cycles (0, 1 … n – 1) of length n, then the corresponding graphs are the n-vertex circulant graphs.In this paper we consider transitive permutation groups of degree mn generated by a “rotation” ρ which is a product of m disjoint cycles of length n and by a “twisted translation” t; such that τρτ–l = ρα for some α.

scholarly journals Vertex-transitive graphs which are not Cayley graphs, I

1994 ◽  
Vol 56 (1) ◽  
pp. 53-63 ◽  
Author(s):  
Brendan D. McKay ◽  
Cheryl E. Praeger
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Petersen Graph ◽  
Fourth Power ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractThe Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

2008 ◽  
Vol 15 (03) ◽  
pp. 379-390 ◽  
Author(s):  
Xuesong Ma ◽  
Ruji Wang
Keyword(s):  
Automorphism Group ◽  
Bipartite Graphs ◽  
Type Ii ◽  
Symmetric Graph ◽  
Trivalent Graph ◽  
Block Graphs ◽  
Group A ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

Constructing the vertex-transitive graphs on 24 vertices

1991 ◽  
Vol 25 (1) ◽  
pp. 56-59
Author(s):  
Gordon F. Royle
Keyword(s):  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

scholarly journals On Hamiltonicity of Vertex-Transitive Graphs and Digraphs of Orderp4

1998 ◽  
Vol 72 (1) ◽  
pp. 110-121 ◽  
Author(s):  
Yu Qing Chen
Keyword(s):  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs
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