139 research outputs found
Arc-transitive graphs of valency 8 have a semiregular automorphism
One version of the polycirculant conjecture states that every
vertex-transitive graph has a semiregular automorphism. We give a proof of the
conjecture in the arc-transitive case for graphs of valency 8, which was the
smallest open case.Comment: 5 page
Semiregular automorphisms of edge-transitive graphs
The polycirculant conjecture asserts that every vertex-transitive digraph has
a semiregular automorphism, that is, a nontrivial automorphism whose cycles all
have the same length. In this paper we investigate the existence of semiregular
automorphisms of edge-transitive graphs. In particular, we show that any
regular edge-transitive graph of valency three or four has a semiregular
automorphism
Semiregular automorphisms of cubic vertex-transitive graphs
We characterise connected cubic graphs admitting a vertex- transitive group
of automorphisms with an abelian normal subgroup that is not semiregular. We
illustrate the utility of this result by using it to prove that the order of a
semiregular subgroup of maximum order in a vertex-transitive group of
automorphisms of a connected cubic graph grows with the order of the graph.Comment: This paper settles Problem 6.3 in P. Cameron, M. Giudici, G. Jones,
W. Kantor, M. Klin, D. Maru\v{s}i\v{c}, L. A. Nowitz, Transitive permutation
groups without semiregular subgroups, J. London Math. Soc. 66 (2002),
325--33
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
Elusive groups of automorphisms of digraphs of small valency
A transitive permutation group is called elusive if it contains no
semiregular element. We show that no group of automorphisms of a connected
graph of valency at most four is elusive and determine all the elusive groups
of automorphisms of connected digraphs of out-valency at most three.Comment: To appear in the European Journal of Combinatoric
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
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
Cubic arc-transitive -circulants
For an integer , a graph is called a -circulant if its
automorphism group contains a cyclic semiregular subgroup with orbits on
the vertices. We show that, if is even, there exist infinitely many cubic
arc-transitive -circulants. We conjecture that, if is odd, then a cubic
arc-transitive -circulant has order at most . Our main result is a
proof of this conjecture when is squarefree and coprime to
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
On edge-primitive and 2-arc-transitive graphs
A graph is edge-primitive if its automorphism group acts primitively on the
edge set. In this short paper, we prove that a finite 2-arc-transitive
edge-primitive graph has almost simple automorphism group if it is neither a
cycle nor a complete bipartite graph. We also present two examples of such
graphs, which are 3-arc-transitive and have faithful vertex-stabilizers
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