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 $p$.

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 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.

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).

SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS NEED NOT BE CAYLEY GRAPHS

2001 ◽  
Vol 33 (6) ◽  
pp. 653-661 ◽  
Author(s):  
CAI HENG LI ◽  
CHERYL E. PRAEGER
Keyword(s):  
Wreath Product ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Permutation Groups ◽  
General Study ◽  
Product Action ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

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.

scholarly journals On Isomorphisms of Vertex-transitive Graphs

10.37236/5651 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jing Chen ◽  
Binzhou Xia
Keyword(s):  
Cayley Graphs ◽  
Isomorphism Problem ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study for GI and DGI-groups, defined analogously to the concept of CI and DCI-groups.

On non-Cayley vertex-transitive graphs and the meta-Cayley graphs

2011 ◽  
Vol 34 (4) ◽  
pp. 425-431 ◽  
Author(s):  
Eric Mwambene
Keyword(s):  
Cayley Graphs ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

scholarly journals ON THE STRUCTURE OF GRAPHS WHICH ARE LOCALLY INDISTINGUISHABLE FROM A LATTICE

2016 ◽  
Vol 4 ◽  
Author(s):  
ITAI BENJAMINI ◽  
DAVID ELLIS
Keyword(s):  
Group Theory ◽  
Algebraic Structure ◽  
Klein Bottle ◽  
Discrete Math ◽  
Crystallographic Groups ◽  
Grid Graphs ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

For each integer $d\geqslant 3$, we obtain a characterization of all graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius 3 in $\mathbb{L}^{d}$, the graph of the $d$-dimensional integer lattice. The finite, connected graphs with this property have a highly rigid, ‘global’ algebraic structure; they can be viewed as quotient lattices of $\mathbb{L}^{d}$ in various compact $d$-dimensional orbifolds which arise from crystallographic groups. We give examples showing that ‘radius 3’ cannot be replaced by ‘radius 2’, and that ‘orbifold’ cannot be replaced by ‘manifold’. In the $d=2$ case, our methods yield new proofs of structure theorems of Thomassen [‘Tilings of the Torus and Klein bottle and vertex-transitive graphs on a fixed surface’, Trans. Amer. Math. Soc.323 (1991), 605–635] and of Márquez et al. [‘Locally grid graphs: classification and Tutte uniqueness’, Discrete Math.266 (2003), 327–352], and also yield short, ‘algebraic’ restatements of these theorems. Our proofs use a mixture of techniques and results from combinatorics, geometry and group theory.

scholarly journals A Note on Constructing Large Cayley Graphs of Given Degree and Diameter by Voltage Assignments

10.37236/1347 ◽  
1997 ◽  
Vol 5 (1) ◽  
Author(s):  
Ljiljana Branković ◽  
Mirka Miller ◽  
Ján Plesník ◽  
Joe Ryan ◽  
Jozef Širáň
Keyword(s):  
Cayley Graphs ◽  
Computer Search ◽  
Large Graphs ◽  
Covering Spaces ◽  
Semidirect Products ◽  
Cyclic Groups ◽  
Small Base ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree $\le 15$ and diameter $\le 10$ have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and diameter.

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

10.37236/4842 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Joy Morris ◽  
Pablo Spiga ◽  
Gabriel Verret
Keyword(s):  
Transitive Group ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Maximum Order ◽  
Abelian Normal Subgroup ◽  
Group Of Automorphisms ◽  
Normal Quotient ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

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.

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