scholarly journals A Family of Half-Transitive Graphs

10.37236/3140 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Jing Chen ◽  
Cai Heng Li ◽  
Ákos Seress
Keyword(s):  
Cayley Graphs ◽  
Infinite Family ◽  
Transitive Graphs

We construct an infinite family of half-transitive graphs, which contains infinitely many Cayley graphs, and infinitely many non-Cayley graphs.

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 An infinite family of cubic nonnormal Cayley graphs on nonabelian simple groups

2018 ◽  
Vol 341 (5) ◽  
pp. 1282-1293
Author(s):  
Jiyong Chen ◽  
Binzhou Xia ◽  
Jin-Xin Zhou
Keyword(s):  
Cayley Graphs ◽  
Infinite Family ◽  
Simple Groups

scholarly journals Vertex-primitive half-transitive graphs

1994 ◽  
Vol 57 (1) ◽  
pp. 113-124 ◽  
Author(s):  
D. E. Taylor ◽  
Ming-Yao Xu
Keyword(s):  
Prime Number ◽  
Infinite Family ◽  
Primitive Groups ◽  
Almost All ◽  
Transitive Graphs ◽  
Vertex Primitive

AbstractGiven an infinite family of finite primitive groups, conditions are found which ensure that almost all the orbitals are not self-paired. If p is a prime number congruent to ±1(mod 10), these conditions apply to the groups P S L (2, p) acting on the cosets of a subgroup isomorphic to A5. In this way, infinitely many vertex-primitive ½-transitive graphs which are not metacirculants are obtained.

scholarly journals Constructing graphs which are ½-transitive

1994 ◽  
Vol 56 (3) ◽  
pp. 391-402 ◽  
Author(s):  
Brian Alspach ◽  
Dragan Marušič ◽  
Lewis Nowitz
Keyword(s):  
Infinite Family ◽  
Edge Transitive ◽  
Transitive Graphs

AbstractAn infinite family of vertex-and edge-transitive, but not arc-transitive, graphs of degree 4 is constructed.

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 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.

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.

Arc-Transitive Graphs of Square-free Order with Valency 11

2021 ◽  
Vol 28 (04) ◽  
pp. 645-654
Author(s):  
Guang Li ◽  
Bo Ling ◽  
Zaiping Lu
Keyword(s):  
Simple Group ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Cayley Graphs ◽  
Complete List ◽  
Dihedral Groups ◽  
Transitive Graphs ◽  
Free Order ◽  
Normal Cayley Graphs

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

scholarly journals On Certain Edge-Transitive Bicirculants

10.37236/7588 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Robert Jajcay ◽  
Štefko Miklavič ◽  
Primož Šparl ◽  
Gorazd Vasiljević
Keyword(s):  
Infinite Family ◽  
Main Theme ◽  
Induced Subgraphs ◽  
Generalized Petersen Graphs ◽  
Symmetry Properties ◽  
Petersen Graphs ◽  
Even Order ◽  
Edge Transitive ◽  
Transitive Graphs

A graph $\Gamma$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. For valences $3$, $4$ and $5$, where the corresponding bicirculants are called generalized Petersen graphs, Rose window graphs and Tabač jn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Tabač jn graphs, infinite families of edge-transitive Rose window graphs exist. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified. Moreover, the classification of the ones of valence $6$ and girth $3$ is given. As a corollary, an infinite family of half-arc-transitive graphs of valence $6$ with universal reachability relation, which were thus far not known to exist, is obtained.

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