443 research outputs found
On vertex-uniprimitive non-Cayley graphs of order pq
Let and be distinct odd primes. Let
be a non-Cayley vertex-transitive graph of order Let G\leq \Aut(\Gamma)
acts primitively on the vertex set . In this paper, we show that
is uniprimitive which is primitive but not 2-transitive and we obtain some
information about and the minimality of the Socle T=\soc(G).Comment: 5 page
Cayley numbers with arbitrarily many distinct prime factors
A positive integer is a Cayley number if every vertex-transitive graph of
order is a Cayley graph. In 1983, Dragan Maru\v{s}i\v{c} posed the problem
of determining the Cayley numbers. In this paper we give an infinite set of
primes such that every finite product of distinct elements from is a Cayley
number. This answers a 1996 outstanding question of Brendan McKay and Cheryl
Praeger, which they "believe to be the key unresolved question" on Cayley
numbers.
We also show that, for every finite product of distinct elements from
, every transitive group of degree contains a semiregular element
Cubic vertex-transitive non-Cayley graphs of order 12p
A graph is said to be {\em vertex-transitive non-Cayley} if its full
automorphism group acts transitively on its vertices and contains no subgroups
acting regularly on its vertices. In this paper, a complete classification of
cubic vertex-transitive non-Cayley graphs of order , where is a prime,
is given. As a result, there are sporadic and one infinite family of such
graphs, of which the sporadic ones occur when , or , and the
infinite family exists if and only if , and in this family
there is a unique graph for a given order.Comment: This paper has been accepted for publication in SCIENCE CHINA
Mathematic
Eigenvalues of Cayley graphs
We survey some of the known results on eigenvalues of Cayley graphs and their
applications, together with related results on eigenvalues of Cayley digraphs
and generalizations of Cayley graphs
On groups all of whose Haar graphs are Cayley graphs
A Cayley graph of a group is a finite simple graph such that
contains a subgroup isomorphic to acting regularly on
, while a Haar graph of is a finite simple bipartite graph
such that contains a subgroup isomorphic to
acting semiregularly on and the -orbits are equal to the
bipartite sets of . A Cayley graph is a Haar graph exactly when it is
bipartite, but no simple condition is known for a Haar graph to be a Cayley
graph. In this paper, we show that the groups and
are the only finite inner abelian groups all of whose Haar graphs are
Cayley graphs (a group is called inner abelian if it is non-abelian, but all of
its proper subgroups are abelian). As an application, it is also shown that
every non-solvable group has a Haar graph which is not a Cayley graph.Comment: 17 page
A classification of tetravalent edge-transitive metacirculants of odd order
In this paper a classification of tetravalent edge-transitive metacirculants
is given. It is shown that a tetravalent edge-transitive metacirculant
is a normal graph except for four known graphs. If further, is a
Cayley graph of a non-abelian metacyclic group, then is
half-transitive
Cayley graphs of diameter two with order greater than 0.684 of the Moore bound for any degree
It is known that the number of vertices of a graph of diameter two cannot
exceed . In this contribution we give a new lower bound for orders of
Cayley graphs of diameter two in the form valid for all
degrees . The result is a significant improvement of currently
known results on the orders of Cayley graphs of diameter two.Comment: 14 pages, 2 tables, Published in European Journal of Combinatorics.
Free access to the article valid until July 9, 2016:
http://authors.elsevier.com/a/1T3zuiVNjvDA
On Isomorphisms of Vertex-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 Color Preserving Automorphisms of Cayley Graphs of Odd Square-free Order
An automorphism of a Cayley graph of a group with
connection set is color-preserving if or
for every edge . If every color-preserving
automorphism of is also affine, then is a CCA (Cayley
color automorphism) graph. If every Cayley graph is a CCA graph,
then is a CCA group. Hujdurovi\'c, Kutnar, D.W. Morris, and J. Morris have
shown that every non-CCA group contains a section isomorphic to the
nonabelian group of order . We first show that there is a unique
non-CCA Cayley graph of . We then show that if is a
non-CCA graph of a group of odd square-free order, then for some CCA group , and .Comment: 16 page
Groups for which it is easy to detect graphical regular representations
We say that a finite group G is "DRR-detecting" if, for every subset S of G,
either the Cayley digraph Cay(G,S) is a digraphical regular representation
(that is, its automorphism group acts regularly on its vertex set) or there is
a nontrivial group automorphism phi of G such that phi(S) = S. We show that
every nilpotent DRR-detecting group is a p-group, but that the wreath product
of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We
also show that if G and H are nontrivial groups that admit a digraphical
regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting,
then the direct product G x H is not DRR-detecting. Some of these results also
have analogues for graphical regular representations.Comment: 11 pages. v2: added acknowledgments and author addres
- …