573 research outputs found
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
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
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
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 basic graphs of symmetric graphs of valency five
A graph \G is {\em symmetric} or {\em arc-transitive} if its automorphism
group \Aut(\G) is transitive on the arc set of the graph, and \G is {\em
basic} if \Aut(\G) has no non-trivial normal subgroup such that the
quotient graph \G_N has the same valency with \G. In this paper, we
classify symmetric basic graphs of order and valency 5, where are
two primes and is a positive integer. It is shown that such a graph is
isomorphic to a family of Cayley graphs on dihedral groups of order with
5\di (q-1), the complete graph of order , the complete bipartite
graph of order 10, or one of the nine sporadic coset graphs
associated with non-abelian simple groups. As an application, connected
pentavalent symmetric graphs of order for some small integers and
are classified
Semisymmetric graphs of order
A simple undirected graph is said to be {\em semisymmetric} if it is regular
and edge-transitive but not vertex-transitive. Every semisymmetric graph is a
bipartite graph with two parts of equal size. It was proved in [{\em J. Combin.
Theory Ser. B} {\bf 3}(1967), 215-232] that there exist no semisymmetric graphs
of order and , where is a prime. The classification of
semisymmetric graphs of order was given in [{\em Comm. in Algebra} {\bf
28}(2000), 2685-2715], for any distinct primes and . Our long term goal
is to determine all the semisymmetric graphs of order , for any prime
. All these graphs \G are divided into two subclasses: (I) \Aut(\G) acts
unfaithfully on at least one bipart; and (II) \Aut(\G) acts faithfully on
both biparts. This paper gives a group theoretical characterization for
Subclass (I) and based on this characterization, we shall give a complete
classification for this subclass in our further research.Comment: 20 page
Automorphisms and Enumeration of Maps of Cayley Graph of a Finite Group
A map is a connected topological graph cellularly embedded in a
surface. In this paper, applying Tutte's algebraic representation of map, new
ideas for enumerating non-equivalent orientable or non-orientable maps of graph
are presented. By determining automorphisms of maps of Cayley graph
with on locally,
orientable and non-orientable surfaces, formulae for the number of
non-equivalent maps of on surfaces (orientable, non-orientable or
locally orientable) are obtained . Meanwhile, using reseults on GRR graph for
finite groups, we enumerate the non-equivalent maps of GRR graph of symmetric
groups, groups generated by 3 involutions and abelian groups on orientable or
non-orientable surfaces.Comment: 17 pages with 1 figur
Nowhere-zero 3-flows in graphs admitting solvable arc-transitive groups of automorphisms
Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a
nowhere-zero 3-flow. In this note we prove that every regular graph of valency
at least four admitting a solvable arc-transitive group of automorphisms admits
a nowhere-zero 3-flow.Comment: This is the final version to be published in: Ars Mathematica
Contemporanea (http://amc-journal.eu/index.php/amc
A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two
A complete list of all connected arc-transitive asymmetric digraphs of
in-valence and out-valence 2 on up to 1000 vertices is presented. As a
byproduct, a complete list of all connected 4-valent graphs admitting a
half-arc-transitive group of automorphisms on up to 1000 vertices is obtained.
Several graph-theoretical properties of the elements of our census are
calculated and discussed
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