681 research outputs found
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
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
Tetravalent edge-transitive Cayley graphs of Frobenius groups
In this paper, we give a characterization for a class of edge-transitive
Cayley graphs, and provide methods for constructing Cayley graphs with certain
symmetry properties. Also this study leads to construct and characterise a new
family of half-transitive graphs
Lifting a prescribed group of automorphisms of graphs
In this paper we are interested in lifting a prescribed group of
automorphisms of a finite graph via regular covering projections. Here we
describe with an example the problems we address and refer to the introductory
section for the correct statements of our results.
Let be the Petersen graph, say, and let be a regular
covering projection. With the current covering machinery, it is straightforward
to find with the property that every subgroup of \Aut(P) lifts via
. However, for constructing peculiar examples and in applications, this is
usually not enough. Sometimes it is important, given a subgroup of
\Aut(P), to find along which lifts but no further automorphism of
does. For instance, in this concrete example, it is interesting to find a
covering of the Petersen graph lifting the alternating group but not the
whole symmetric group . (Recall that \Aut(P)\cong S_5.) Some other time
it is important, given a subgroup of \Aut(P), to find with the
property that \Aut(\tilde{P}) is the lift of . Typically, it is desirable
to find satisfying both conditions. In a very broad sense, this might
remind wallpaper patterns on surfaces: the group of symmetries of the
dodecahedron is , and there is a nice colouring of the dodecahedron (found
also by Escher) whose group of symmetries is just .
In this paper, we address this problem in full generality.Comment: 10 page
On the Saxl graph of a permutation group
Let be a permutation group on a set . A subset of is a
base for if its pointwise stabiliser in is trivial. In this paper we
introduce and study an associated graph , which we call the Saxl
graph of . The vertices of are the points of , and two
vertices are adjacent if they form a base for . This graph encodes some
interesting properties of the permutation group. We investigate the
connectivity of for a finite transitive group , as well as its
diameter, Hamiltonicity, clique and independence numbers, and we present
several open problems. For instance, we conjecture that if is a primitive
group with a base of size , then the diameter of is at most .
Using a probabilistic approach, we establish the conjecture for some families
of almost simple groups. For example, the conjecture holds when or
(with ) and the point stabiliser of is a primitive subgroup. In
contrast, we can construct imprimitive groups whose Saxl graph is disconnected
with arbitrarily many connected components, or connected with arbitrarily large
diameter.Comment: 27 pages; to appear in Math. Proc. Cambridge Philos. So
Normal Edge-Transitive Cayley Graphs of Frobenius Groups
A Cayley Graph for a group is called normal edge-transitive if it admits
an edge-transitive action of some subgroup of the Holomorph of (the
normaliser of a regular copy of in ). We complete
the classification of normal edge-transitive Cayley graphs of order a product
of two primes by dealing with Cayley graphs for Frobenius groups of such
orders. We determine the automorphism groups of these graphs, proving in
particular that there is a unique vertex-primitive example, namely the flag
graph of the Fano plane
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
Finite 2-geodesic transitive graphs of prime valency
We classify non-complete prime valency graphs satisfying the property that
their automorphism group is transitive on both the set of arcs and the set of
-geodesics. We prove that either is 2-arc transitive or the valency
satisfies , and for each such prime there is a unique
graph with this property: it is a non-bipartite antipodal double cover of the
complete graph with automorphism group and
diameter 3.Comment: arXiv admin note: substantial text overlap with arXiv:1110.223
A classification of tetravalent edge-transitive metacirculants of odd order
In this paper a classification of tetravalent edge-transitive metacirculants
is given. It is shown that a tetravalent edge-transitive metacirculant
is a normal graph except for four known graphs. If further, is a
Cayley graph of a non-abelian metacyclic group, then is
half-transitive
On vertex stabilisers in symmetric quintic graphs
In this paper we determine all locally finite and symmetric actions of a
group on the tree of valency five. As a corollary we complete the
classification of the isomorphism types of vertex and edge stabilisers in a
group acting symmetrically on a graph of valency five. This builds on work of
Weiss and recent work of Zhou and Feng. This depends upon the second result of
this paper, the classification of isomorphism types of finite, primitive
amalgams of degree (5, 2)
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