Let Γ be a finite simple undirected graph and G ≤ Aut(Γ). If G is transitive on the set of s-arcs but not on the set of (s+1)-arcs of Γ, then Γ is called (G, s)-transitive. For a connected (G, ...s)-transitive graph Γ of prime valency, the vertex-stabilizer Gα with α ∈ V(Γ) has been determined when Gα is solvable. In this paper, we give a characterization of the vertex-stabilizers of (G, s)-transitive graphs of prime valency when Gα is unsolvable.
It is known that arc-transitive group actions on finite cubic (3-valent) graphs fall into seven classes, denoted by 1, 21, 22, 3, 41, 42 and 5, where k, k1 or k2 indicates that the action is ...k-arc-regular, with k2 indicating that there is no arc-reversing automorphism of order 2 (for k=2 or 4). These classes can be further subdivided into 17 sub-classes, according to the types of arc-transitive subgroups of the full automorphism group of the graph, sometimes called the ‘action type’ of the graph. In this paper, we complete the determination of the smallest graphs in each of these 17 classes (begun by Conder and Nedela in (2009) 7).
This paper is one of a series of papers devoted to characterizing edge-transitive graphs of square-free order. It presents a complete list of locally-primitive arc-transitive graphs of square-free ...order and valency d∈{5,6,7}.
In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let Γ be such a Cayley graph on a non-abelian simple group T. It is ...proved that either Γ is a normal Cayley graph or Γ is S-arc-transitive, with (S,T)=(An,An−1) and n=7,21,63 or 84; and, for each of these four values of n, there really exist some arc-transitive 7-valent non-normal Cayley graphs on An−1 and specific examples are constructed.
The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, ...which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition \(c_2=1\) (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with \(c_2=1\) is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine \(2\)-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with \(c_2=1\) that admit an automorphism group acting \(2\)-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with \(c_2=1\) and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.
Let
p
be a prime, and let
Λ
2
p
be a connected cubic arc-transitive graph of order 2
p
. In the literature, a lot of works have been done on the classification of edge-transitive normal covers of
Λ
2
...p
for specific
p
≤
7
. An interesting problem is to generalize these results to an arbitrary prime
p
. In 2014, Zhou and Feng classified edge-transitive cyclic or dihedral normal covers of
Λ
2
p
for each prime
p
. In our previous work, we classified all edge-transitive
N
-normal covers of
Λ
2
p
, where
p
is a prime and
N
is a metacyclic 2-group. In this paper, we give a classification of edge-transitive
N
-normal covers of
Λ
2
p
, where
p
≥
5
is a prime and
N
is a metacyclic group of odd prime power order.
Quite a lot of attention has been paid recently to the construction of edge- or arc-transitive covers of symmetric graphs. In most cases, the approach has involved voltage graph techniques, which are ...excellent for finding regular covers in which the group of covering transformations is either cyclic or elementary abelian, or more generally, homocyclic, but are not so easy to use when the covering group has other forms — even when it is abelian but not homocyclic. In this paper, a different approach is introduced that can be used more widely. This new approach takes a universal group for the action of the automorphism group of the base graph, and uses Reidemeister–Schreier theory to obtain a presentation for a ‘universal covering group’, and some representation theory and other methods for determining suitable quotients. This approach is then used to find all arc-transitive abelian regular covers of K4, K3,3, the cube Q3, and the Petersen graph. A sequel will do the same for the Heawood graph.
A family of tetravalent half-arc-transitive graphs Biswas, Sucharita; Das, Angsuman
Proceedings of the Indian Academy of Sciences. Mathematical sciences,
10/2021, Letnik:
131, Številka:
2
Journal Article
Recenzirano
Alspach
et al
. (
J. Austral. Math. Soc
.
56(3)
(1994) 391–402) constructed an infinite family of tetravalent graphs
M
(
a
;
m
,
n
) and proved that if
n
≥
9
be odd and
a
3
≡
1
(
mod
n
)
, then
M
(
...a
; 3,
n
) is half-arc-transitive. In this paper, we show that if
a
3
≡
1
(
mod
n
)
, then
M
(
a
; 3,
n
) is an infinite family of tetravalent half-arc-transitive Cayley graphs for all integers
n
except 7 and 14.
A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in ...determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.
A classification of connected vertex‐transitive cubic graphs of square‐free order is provided. It is shown that such graphs are well‐characterized metacirculants (including dihedrants, generalized ...Petersen graphs, Möbius bands), or Tutte's 8‐cage, or graphs arisen from simple groups PSL(2, p).
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