228 research outputs found
Every finite group has a normal bi-Cayley graph
A graph \G with a group of automorphisms acting semiregularly on the
vertices with two orbits is called a {\em bi-Cayley graph} over . When
is a normal subgroup of \Aut(\G), we say that \G is {\em normal} with
respect to . In this paper, we show that every finite group has a connected
normal bi-Cayley graph. This improves Theorem~5 of [M. Arezoomand, B. Taeri,
Normality of 2-Cayley digraphs, Discrete Math. 338 (2015) 41--47], and provides
a positive answer to the Question of the above paper
On congruence in Z^n and the dimension of a multidimensional circulant
From a generalization to of the concept of congruence we define a
family of regular digraphs or graphs called multidimensional circulants, which
turn out to be Cayley (di)graphs of Abelian groups. This paper is mainly
devoted to show the relationship between the Smith normal form for integral
matrices and the dimensions of such (di)graphs, that is the minimum ranks of
the groups they can arise from. In particular, those 2-step multidimensional
circulants which are circulants, that is Cayley (di)graphs of cyclic groups,
are fully characterized. In addition, a reasoning due to Lawrence is used to
prove that the cartesian product of circulants with equal number of
vertices , a prime, has dimension
Derangement action digraphs and graphs
We study the family of \emph{derangement action digraphs}, which are a
subfamily of the group action graphs introduced in [Fred Annexstein, Marc
Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel
architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any
non-empty set and a non-empty subset of \Der(X), the set of
derangments of , we define the derangement action digraph
to have vertex set , and an arc from to
if and only if for some . In common with Cayley graphs and
digraphs, derangement action digraphs may be useful to model networks as the
same routing and communication scheme can be implemented at each vertex. We
determine necessary and sufficient conditions on under which
may be viewed as a simple graph of valency ,
and we call such graphs derangement action graphs. Also we investigate the
structural and symmetry properties of these digraphs and graphs. Several open
problems are posed and many examples are given.Comment: 15 pages, 1 figur
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
Automorphism groups of circulant graphs -- a survey
A circulant (di)graph is a (di)graph on n vertices that admits a cyclic
automorphism of order n. This paper provides a survey of the work that has been
done on finding the automorphism groups of circulant (di)graphs, including the
generalisation in which the edges of the (di)graph have been assigned colours
that are invariant under the aforementioned cyclic automorphism.Comment: 16 pages, 0 figures, LaTeX fil
A Generalisation of Isomorphisms with Applications
In this paper, we study the behaviour of TF-isomorphisms, a natural
generalisation of isomorphisms. TF-isomorphisms allow us to simplify the
approach to seemingly unrelated problems. In particular, we mention the
Neighbourhood Reconstruction problem, the Matrix Symmetrization problem and
Stability of Graphs. We start with a study of invariance under TF-isomorphisms.
In particular, we show that alternating trails and incidence double covers are
conserved by TF-isomorphisms, irrespective of whether they are TF-isomorphisms
between graphs or digraphs. We then define an equivalence relation and
subsequently relate its equivalence classes to the incidence double cover of a
graph. By directing the edges of an incidence double cover from one colour
class to the other and discarding isolated vertices we obtain an invariant
under TF-isomorphisms which gathers a number of invariants. This can be used to
study TF-orbitals, an analogous generalisation of the orbitals of a permutation
group.Comment: 27 pages, 8 figure
Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups
Let be a Cayley digraph on a group and let
. The Cayley index of is . It has
previously been shown that, if is a prime, is a cyclic -group and
contains a noncyclic regular subgroup, then the Cayley index of is
superexponential in .
We present evidence suggesting that cyclic groups are exceptional in this
respect. Specifically, we establish the contrasting result that, if is an
odd prime and is abelian but not cyclic, and has order a power of at
least , then there is a Cayley digraph on whose Cayley index
is just , and whose automorphism group contains a nonabelian regular
subgroup
On the Normality of Some Cayley Digraphs with Valency 2
We call a Cayley digraph =Cay(G; S) normal for G if R(G), the rightregular representation of G, is a normal subgroup of the full automorphism groupAut() of . In this paper we determine the normality of Cayley digraphs of valency2 on the groups of order pq and also on non-abelian nite groups G such that everyproper subgroup of G is abelian.DOI : http://dx.doi.org/10.22342/jims.17.2.3.67-7
Automorphism group of the complete alternating group graph
Let and denote the symmetric group and alternating group of
degree with , respectively. Let be the set of all -cycles
in . The \emph{complete alternating group graph}, denoted by , is
defined as the Cayley graph on with respect to .
In this paper, we show that () is not a normal Cayley graph.
Furthermore, the automorphism group of for is obtained, which
equals to , where is the
right regular representation of , is the inner
automorphism group of , and , where is
the map ().Comment: 9 pages, 1 figur
Isomorphisms of Cayley graphs on nilpotent groups
Let S be a finite generating set of a torsion-free, nilpotent group G. We
show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is,
every automorphism of the graph is obtained by composing a group automorphism
with multiplication by an element of the group.) More generally, we show that
if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two
nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S')
factors through to a well-defined affine map from G/N to G'/N', where N and N'
are the torsion subgroups of G and G', respectively. For the special case where
the groups are abelian, these results were previously proved by A.A.Ryabchenko
and C.Loeh, respectively.Comment: 12 pages, plus 7 pages of notes to aid the referee. One of our
corollaries is already known, so a reference to the literature has been adde
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