2,066 research outputs found
Half-arc-transitive graphs of prime-cube order of small valencies
A graph is called {\em half-arc-transitive} if its full automorphism group
acts transitively on vertices and edges, but not on arcs. It is well known that
for any prime there is no tetravalent half-arc-transitive graph of order
or . Xu~[Half-transitive graphs of prime-cube order, J. Algebraic
Combin. 1 (1992) 275-282] classified half-arc-transitive graphs of order
and valency . In this paper we classify half-arc-transitive graphs of order
and valency or . In particular, the first known infinite family of
half-arc-transitive Cayley graphs on non-metacyclic -groups is constructed.Comment: 13 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
Pentavalent symmetric graphs admitting transitive non-abelian characteristically simple groups
Let be a graph and let be a group of automorphisms of .
The graph is called -normal if is normal in the automorphism
group of . Let be a finite non-abelian simple group and let with . In this paper we prove that if every connected pentavalent
symmetric -vertex-transitive graph is -normal, then every connected
pentavalent symmetric -vertex-transitive graph is -normal. This result,
among others, implies that every connected pentavalent symmetric
-vertex-transitive graph is -normal except is one of simple
groups. Furthermore, every connected pentavalent symmetric -regular graph is
-normal except is one of simple groups, and every connected
pentavalent -symmetric graph is -normal except is one of simple
groups.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1701.0118
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 nonlinear tracking algorithm with range-rate measurements based on unbiased measurement conversion
The three-dimensional CMKF-U with only position measurements is extended to
solve the nonlinear tracking problem with range-rate measurements in this
paper. A pseudo measurement is constructed by the product of range and
range-rate measurements to reduce the high nonlinearity of the range-rate
measurements with respect to the target state; then the mean and covariance of
the converted measurement errors are derived by the measurement conditioned
method, showing better consistency than the transitional nested conditioning
method; finally, the sequential filter was used to process the converted
position and range-rate measurements sequentially to reduce the approximation
error in the second-order EKF. Monte Carlo simulations show that the
performance of the new tracking algorithm is better than the traditional one
based on CMKF-D
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
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
Heptavalent symmetric graphs with solvable stabilizers admitting vertex-transitive non-abelian simple groups
A graph is said to be symmetric if its automorphism group acts transitively on the arc set of . In this paper, we
show that if is a finite connected heptavalent symmetric graph with
solvable stabilizer admitting a vertex-transitive non-abelian simple group
of automorphisms, then either is normal in , or contains a non-abelian simple normal subgroup such that and is explicitly given as one of possible exception pairs of
non-abelian simple groups. Furthermore, if is regular on the vertex set of
then the exception pair is one of possible pairs, and if
is arc-transitive then the exception pair or
.Comment: 9. arXiv admin note: substantial text overlap with arXiv:1701.0118
Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime
A regular cover of a connected graph is called {\em cyclic} or {\em dihedral}
if its transformation group is cyclic or dihedral respectively, and {\em
arc-transitive} (or {\em symmetric}) if the fibre-preserving automorphism
subgroup acts arc-transitively on the regular cover. In this paper, we give a
classification of arc-transitive cyclic and dihedral covers of a connected
pentavalent symmetric graph of order twice a prime. All those covers are
explicitly constructed as Cayley graphs on some groups, and their full
automorphism groups are determined
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
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