Two-arc-transitive bicirculants Jin, Wei
Journal of combinatorial theory. Series B,
November 2023, 2023-11-00, Letnik:
163
Journal Article
Recenzirano
In this paper, we determine the class of finite 2-arc-transitive bicirculants. We show that a connected 2-arc-transitive bicirculant is one of the following graphs: C2n where n⩾2, K2n where n⩾2, Kn,n ...where n⩾3, Kn,n−nK2 where n⩾4, B(PG(d−1,q)) and B′(PG(d−1,q)) where d≥3 and q is a prime power, X1(4,q) where q≡3(mod4) is a prime power, Kq+12d where q is an odd prime power and d≥2 dividing q−1, ATQ(1+q,2d) where d|q−1 and d∤12(q−1), ATD(1+q,2d) where d|12(q−1) and d≥2, Γ(d,q,r), where d≥2, q is a prime power and r|q−1, Petersen graph, Desargues graph, dodecahedron graph, folded 5-cube, X(2,2), X′(3,2), X2(3), ATQ(4,12), GP(12,5), GP(24,5), B(H(11)), B′(H(11)), ATD(4,6) and ATD(5,6).
A finite simple graph
Γ
is called a Nest graph if it is regular of valency 6 and admits an automorphism with two orbits of the same length such that at least one of the subgraphs induced by these ...orbits is a cycle. In this paper, we complete classification of the edge-transitive Nest graphs and by this solve the problem posed by Jajcay et al. (Electron J Comb 26:#P2.6, 2019).
A graph-theoretic environment is used to study the connection between imprimitivity and semiregularity, two concepts arising naturally in the context of permutation groups. Among other, it is shown ...that a connected arc-transitive graph admitting a nontrivial automorphism with two orbits of odd length, together with an imprimitivity block system consisting of blocks of size 2, orthogonal to these two orbits, is either the canonical double cover of an arc-transitive circulant or the wreath product of an arc-transitive circulant with the empty graph
K
¯
2
on two vertices.
The well-known Petersen graph
G
(
5
,
2
)
admits a semi-regular automorphism
α
acting on the vertex set with two orbits of equal size. This makes it a
bicirculant. It is shown that trivalent ...bicirculants fall into four classes. Some basic properties of trivalent bicirculants are explored and the connection to combinatorial and geometric configurations are studied. Some analogues of the polycirculant conjecture are mentioned.
A
bicirculant
is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. A graph is said to be
arc-transitive
if its automorphism group acts transitively on the ...set of its arcs. All cubic and tetravalent arc-transitive bicirculants are known, and this paper gives a complete classification of connected pentavalent arc-transitive bicirculants. In particular, it is shown that, with the exception of seven particular graphs, a connected pentavalent bicirculant is arc-transitive if and only if it is isomorphic to a Cayley graph
Cay
(
D
2
n
,
{
b
,
b
a
,
b
a
r
+
1
,
b
a
r
2
+
r
+
1
,
b
a
r
3
+
r
2
+
r
+
1
}
)
on the dihedral group
D
2
n
=
⟨
a
,
b
∣
a
n
=
b
2
=
b
a
b
a
=
1
⟩
, where
r
∈
Z
n
∗
such that
r
4
+
r
3
+
r
2
+
r
+
1
≡
0
(
mod
n
)
.
A
bicirculant
is a graph admitting an automorphism whose cyclic decomposition consists of two cycles of equal length. In this paper we introduce the
Tabačjn graphs
, a family of pentavalent ...bicirculants which are a natural generalization of generalized Petersen graphs obtained from them by adding two additional perfect matchings between the two orbits of a semiregular automorphism. The main result is the classification of symmetric Tabačjn graphs. In particular, it is shown that the only such graphs are the complete graph
K
6
, the complete bipartite graph minus a perfect matching
K
6
,
6
-
6
K
2
and the icosahedron graph.
We define a class of digraphs involving differences in a group that generalizes Cayley digraphs, that we call difference digraphs. We define also a new combinatorial structure, called partial sum ...family, or PSF for short, from which we obtain difference digraphs that are directed strongly regular graphs. We give an infinite family of PSFs and we give also twelve sporadic ones that generate directed strongly regular graphs whose existence was previously undecided.
An
n-
bicirculant is a graph having an automorphism with two orbits of length
n and no other orbits. Symmetry properties of
p-bicirculants,
p a prime, are extensively studied. In particular, the ...actions of their automorphism groups are described in detail in terms of certain algebraic representation of such graphs.
Abicirculantis a graph admitting an automorphism whose cyclic decomposition consists of two cycles of equal length. In this paper we consider automorphisms of the so-calledTabačjn graphs, a family of ...pentavalent bicirculants which are obtained from the generalized Petersen graphs by adding two additional perfect matchings between the two orbits of the above mentioned automorphism. As a corollary, we determine which Tabačjn graphs are vertex-transitive.
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