488 research outputs found
On 2-Fold Covers of Graphs
A regular covering projection \p\colon \tX \to X of connected graphs is
-admissible if lifts along \p. Denote by \tG the lifted group, and
let \CT(\p) be the group of covering transformations. The projection is
called -split whenever the extension \CT(\p) \to \tG \to G splits. In this
paper, split 2-covers are considered. Supposing that is transitive on ,
a -split cover is said to be -split-transitive if all complements \bG
\cong G of \CT(\p) within \tG are transitive on \tX; it is said to be
-split-sectional whenever for each complement \bG there exists a
\bG-invariant section of \p; and it is called -split-mixed otherwise.
It is shown, when is an arc-transitive group, split-sectional and
split-mixed 2-covers lead to canonical double covers. For cubic symmetric
graphs split 2-cover are necessarily cannonical double covers when is 1- or
4-regular. In all other cases, that is, if is -regular, or 5, a
necessary and sufficient condition for the existence of a transitive complement
\bG is given, and an infinite family of split-transitive 2-covers based on
the alternating groups of the form is constructed.
Finally, chains of consecutive 2-covers, along which an arc-transitive group
has successive lifts, are also considered. It is proved that in such a
chain, at most two projections can be split. Further, it is shown that, in the
context of cubic symmetric graphs, if exactly two of them are split, then one
is split-transitive and the other one is either split-sectional or split-mixed.Comment: 18 pages, 3 figure
On cubic symmetric non-Cayley graphs with solvable automorphism groups
It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs
with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015),
1-11] that a cubic symmetric graph with a solvable automorphism group is either
a Cayley graph or a -regular graph of type , that is, a graph with no
automorphism of order interchanging two adjacent vertices. In this paper an
infinite family of non-Cayley cubic -regular graphs of type with a
solvable automorphism group is constructed. The smallest graph in this family
has order 6174.Comment: 8 page
Permutation groups and normal subgroups
Various descending chains of subgroups of a finite permutation group can be
used to define a sequence of `basic' permutation groups that are analogues of
composition factors for abstract finite groups. Primitive groups have been the
traditional choice for this purpose, but some combinatorial applications
require different kinds of basic groups, such as quasiprimitive groups, that
are defined by properties of their normal subgroups. Quasiprimitive groups
admit similar analyses to primitive groups, share many of their properties, and
have been used successfully, for example to study -arc transitive graphs.
Moreover investigating them has led to new results about finite simple groups
On the orders of arc-transitive graphs
A graph is called {\em arc-transitive} (or {\em symmetric}) if its
automorphism group has a single orbit on ordered pairs of adjacent vertices,
and 2-arc-transitive its automorphism group has a single orbit on ordered paths
of length 2. In this paper we consider the orders of such graphs, for given
valency. We prove that for any given positive integer , there exist only
finitely many connected 3-valent 2-arc-transitive graphs whose order is
for some prime , and that if , then there exist only finitely many
connected -valent 2-arc-transitive graphs whose order is or for
some prime . We also prove that there are infinitely many (even) values of
for which there are only finitely many connected 3-valent symmetric graphs
of order where is prime
Constructing 2-Arc-Transitive Covers of Hypercubes
We introduce the notion of a symmetric basis of a vector space equipped with
a quadratic form, and provide a sufficient and necessary condition for the
existence to such a basis. Symmetric bases are then used to study Cayley graphs
of certain extraspecial 2-groups of order 2^{2r+1} (r\geq 1), which are further
shown to be normal Cayley graphs and 2-arc-transitive covers of 2r-dimensional
hypercubes
Semisymmetric elementary abelian covers of the M\"obius-Kantor graph
Let \p_N \colon \tX \to X be a regular covering projection of connected
graphs with the group of covering transformations isomorphic to . If is
an elementary abelian -group, then the projection \p_N is called
-elementary abelian. The projection \p_N is vertex-transitive
(edge-transitive) if some vertex-transitive (edge-transitive) subgroup of \Aut
X lifts along \p_N, and semisymmetric if it is edge- but not
vertex-transitive. The projection \p_N is minimal semisymmetric if
cannot be written as a composition \p_N = \p \circ \p_M of two (nontrivial)
regular covering projections, where \p_M is semisymmetric.
Finding elementary abelian covering projections can be grasped
combinatorially via a linear representation of automorphisms acting on the
first homology group of the graph. The method essentially reduces to finding
invariant subspaces of matrix groups over prime fields (see {\em J. Algebr.
Combin.}, {\bf 20} (2004), 71--97).
In this paper, all pairwise nonisomorphic minimal semisymmetric elementary
abelian regular covering projections of the M\"{o}bius-Kantor graph, the
Generalized Petersen graph \GP(8,3), are constructed. No such covers exist
for . Otherwise, the number of such covering projections is equal to
and in cases and , respectively, and to and in cases
and , respectively.
For each such covering projection the voltage rules generating the
corresponding covers are displayed explicitly.Comment: 23 pages, 8 figure
Four-Valent Oriented Graphs of Biquasiprimitive Type
Let denote the family of all graph-group pairs
where is 4-valent, connected and -oriented
(-half-arc-transitive). Using a novel application of the structure theorem
for biquasiprimitive permutation groups of the second author, we produce a
description of all pairs for which every
nontrivial normal subgroup of has at most two orbits on the vertices of
. In particular we show that has a unique minimal normal subgroup
and that for a simple group and . This
provides a crucial step towards a general description of the long-studied
family in terms of a normal quotient reduction. We also give
several methods for constructing pairs of this type and provide
many new infinite families of examples, covering each of the possible
structures of the normal subgroup
Graphs that contain multiply transitive matchings
Let be a finite, undirected, connected, simple graph. We say that a
matching is a \textit{permutable -matching} if
contains edges and the subgroup of that fixes the
matching setwise allows the edges of to be permuted
in any fashion. A matching is \textit{2-transitive} if the
setwise stabilizer of in can map any ordered
pair of distinct edges of to any other ordered pair of distinct
edges of . We provide constructions of graphs with a permutable
matching; we show that, if is an arc-transitive graph that contains a
permutable -matching for , then the degree of is at least
; and, when is sufficiently large, we characterize the locally
primitive, arc-transitive graphs of degree that contain a permutable
-matching. Finally, we classify the graphs that have a -transitive
perfect matching and also classify graphs that have a permutable perfect
matching.Comment: to appear in European Journal of Combinatoric
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
On quartic half-arc-transitive metacirculants
Following Alspach and Parsons, a {\em metacirculant graph} is a graph
admitting a transitive group generated by two automorphisms and
, where is -semiregular for some integers , , and where normalizes , cyclically permuting the orbits
of in such a way that has at least one fixed vertex. A {\em
half-arc-transitive graph} is a vertex- and edge- but not arc-transitive graph.
In this article quartic half-arc-transitive metacirculants are explored and
their connection to the so called tightly attached quartic half-arc-transitive
graphs is explored. It is shown that there are three essentially different
possibilities for a quartic half-arc-transitive metacirculant which is not
tightly attached to exist. These graphs are extensively studied and some
infinite families of such graphs are constructed.Comment: 31 pages, 2 figure
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