270 research outputs found
Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs
A finite simple graph is called a bi-Cayley graph over a group if it has
a semiregular automorphism group, isomorphic to which has two orbits on
the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups
have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014),
679--693). In this paper we consider the latter class of graphs and select
those in the class which are also arc-transitive. Furthermore, such a graph is
called -type when it is bipartite, and the bipartition classes are equal to
the two orbits of the respective semiregular automorphism group. A -type
graph can be represented as the graph where is a
subset of the vertex set of which consists of two copies of say
and and the edge set is . A
bi-Cayley graph is called a BCI-graph if for any bi-Cayley
graph
implies that for some and . It is also shown that every cubic connected arc-transitive
-type bi-Cayley graph over an abelian group is a BCI-graph
Vertex-transitive Haar graphs that are not Cayley graphs
In a recent paper (arXiv:1505.01475 ) Est\'elyi and Pisanski raised a
question whether there exist vertex-transitive Haar graphs that are not Cayley
graphs. In this note we construct an infinite family of trivalent Haar graphs
that are vertex-transitive but non-Cayley. The smallest example has 40 vertices
and is the well-known Kronecker cover over the dodecahedron graph ,
occurring as the graph in the Foster census of connected symmetric
trivalent graphs.Comment: 9 pages, 2 figure
Edge-transitive bi-Cayley graphs
A graph \G admitting a group of automorphisms acting semi-regularly on
the vertices with exactly two orbits is called a {\em bi-Cayley graph\/} over
. Such a graph \G is called {\em normal\/} if is normal in the full
automorphism group of \G, and {\em normal edge-transitive\/} if the
normaliser of in the full automorphism group of \G is transitive on the
edges of \G. % In this paper, we give a characterisation of normal
edge-transitive bi-Cayley graphs, %which form an important subfamily of
bi-Cayley graphs, and in particular, we give a detailed description of
-arc-transitive normal bi-Cayley graphs. Using this, we investigate three
classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups
and metacyclic -groups. We find that under certain conditions, `normal
edge-transitive' is the same as `normal' for graphs in these three classes. As
a by-product, we obtain a complete classification of all connected trivalent
edge-transitive graphs of girth at most , and answer some open questions
from the literature about -arc-transitive, half-arc-transitive and
semisymmetric graphs
A classification of nilpotent 3-BCI groups
Given a finite group and a subset the bi-Cayley graph
\bcay(G,S) is the graph whose vertex set is and edge set
is . A bi-Cayley graph \bcay(G,S)
is called a BCI-graph if for any bi-Cayley graph \bcay(G,T), \bcay(G,S)
\cong \bcay(G,T) implies that for some and \alpha
\in \aut(G). A group is called an -BCI-group if all bi-Cayley graphs of
of valency at most are BCI-graphs.In this paper we prove that, a finite
nilpotent group is a 3-BCI-group if and only if it is in the form
where is a homocyclic group of odd order, and is trivial or one of the
groups and \Q_8
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
Arc-transitive bicirculants
In this paper, we characterise the family of finite arc-transitive
bicirculants. We show that every finite arc-transitive bicirculant is a normal
-cover of an arc-transitive graph that lies in one of eight infinite
families or is one of seven sporadic arc-transitive graphs. Moreover, each of
these ``basic'' graphs is either an arc-transitive bicirculant or an
arc-transitive circulant, and each graph in the latter case has an
arc-transitive bicirculant normal -cover for some integer
Cubic symmetric graphs having an abelian automorphism group with two orbits
Finite connected cubic symmetric graphs of girth 6 have been classified by K.
Kutnar and D. Maru\v{s}i\v{c}, in particular, each of these graphs has an
abelian automorphism group with two orbits on the vertex set. In this paper all
cubic symmetric graphs with the latter property are determined. In particular,
with the exception of the graphs K_4, K_{3,3}, Q_3, GP(5,2), GP(10,2), F40 and
GP(24,5), all the obtained graphs are of girth 6.Comment: Keywords: cubic symmetric graph, Haar graph, voltage graph. This
papaer has been withdrawn by the author because it is an outdated versio
Searching for Large Circulant Graphs
We address the problem of constructing large undirected circulant networks
with given degree and diameter. First we discuss the theoretical upper bounds
and their asymptotics, and then we describe and implement a computer-based
method to find large circulant graphs with given parameters. For several
combinations of degree and diameter, our algorithm produces the largest known
circulant graphs. We summarize our findings in a table, up to degree 15 and
diameter 10, and we perform a statistical analysis of this table, which can be
useful for evaluating the performance of our methods, as well as other
constructions in the future
Asymptotic enumeration of vertex-transitive graphs of fixed valency
Let be a group and let be an inverse-closed and identity-free
generating set of . The \emph{Cayley graph} \Cay(G,S) has vertex-set
and two vertices and are adjacent if and only if . Let
be the number of isomorphism classes of -valent Cayley graphs of
order at most . We show that , as
. We also obtain some stronger results in the case .Comment: 15 page
Bipartite bi-Cayley graphs over metacyclic groups of odd prime-power order
A graph is a bi-Cayley graph over a group if is a
semiregular group of automorphisms of having two orbits. Let be a
non-abelian metacyclic -group for an odd prime , and let be a
connected bipartite bi-Cayley graph over the group . In this paper, we prove
that is normal in the full automorphism group of
when is a Sylow -subgroup of . As an
application, we classify half-arc-transitive bipartite bi-Cayley graphs over
the group of valency less than . Furthermore, it is shown that there
are no semisymmetric and no arc-transitive bipartite bi-Cayley graphs over the
group of valency less than .Comment: 20 pages, 1 figur
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