A graph is called half-arc-transitive if its full automorphism group acts transitively on vertices and edges, but not on arcs. In this paper, we classify hexavalent half-arc-transitive graphs of ...order 9p for each prime p. As a result, there are four infinite families of such graphs: three defined on Zp⋊Z27 with 27|(p−1); one defined on Z3p⋊Z9 with 9|(p−1).
It is known that there are precisely three transitive permutation groups of degree 6 that admit an invariant partition with three parts of size 2 such that the kernel of the action on the parts has ...order 4; these groups are called
A
4
(
6
),
S
4
(
6
d
) and
S
4
(
6
c
). For each
L
∈
{
A
4
(
6
)
,
S
4
(
6
d
)
,
S
4
(
6
c
)
}, we construct an infinite family of finite connected 6‐valent graphs
{
Γ
n
}
n
∈
N and arc‐transitive groups
G
n
≤
Aut
(
Γ
n
) such that the permutation group induced by the action of the vertex‐stabiliser
(
G
n
)
v on the neighbourhood of a vertex
v is permutation isomorphic to
L, and such that
∣
(
G
n
)
v
∣ is exponential in
∣
V
(
Γ
n
)
∣. These three groups were the only transitive permutation groups of degree at most 7 for which the existence of such a family was undecided. In the process, we construct an infinite family of cubic 2‐arc‐transitive graphs such that the dimension of the 1‐eigenspace over the field of order 2 of the adjacency matrix of the graph grows linearly with the order of the graph.
In the late 1990s, Graver and Watkins initiated the study of all edge‐transitive maps. Recently, Gareth Jones revisited the study of such maps and suggested classifying the maps in terms of either ...their automorphism groups or their underlying graphs. A natural step towards classifying edge‐transitive maps is to study the arc‐transitive ones. In this paper, we investigate the connection of a class of arc‐transitive maps to consistent cycles of the underlying graph, with special emphasis on maps of smallest possible valence, namely 4. We give a complete classification of arc‐transitive maps whose underlying graphs are arc‐transitive Rose Window graphs.
A graph, with a group G of its automorphisms, is said to be (G,s)-transitive if G is transitive on s-arcs but not on (s+1)-arcs of the graph. Let X be a connected (G,s)-transitive graph for some s≥1, ...and let Gv be the stabilizer of a vertex v∈V(X) in G. In this paper, we determine the structure of Gv when X has valency 5 and Gv is non-solvable. Together with the results of Zhou and Feng J.-X. Zhou, Y.-Q. Feng, On symmetric graphs of valency five, Discrete Math. 310 (2010) 1725–1732, the structure of Gv is completely determined when X has valency 5. For valency 3 or 4, the structure of Gv is known.
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 combinatorial graph Γ is symmetric, or arc-transitive, if its automorphism group acts transitively on the arcs of Γ, and s-arc-transitive (resp. s-arc-regular) if its automorphism group acts ...transitively (resp. regularly) on the set of s-arcs of Γ, which are the walks of length s in Γ in which any three consecutive vertices are distinct. It was shown by Tutte (1947, 1959) that every finite symmetric trivalent graph is s-arc-regular for some s≤5. Djoković and Miller (1980) took this further by showing that there are seven types of arc-transitive group action on finite trivalent graphs, characterised by the stabilisers of a vertex and an edge. The latter classification was refined by Conder and Nedela (2009), in terms of what types of arc-transitive subgroup can occur in the automorphism group of Γ.
In this paper we address the question of when a finite trivalent Cayley graph is arc-transitive, by determining when a connected finite arc-transitive trivalent graph is a Cayley graph. We show that in five of the 17 Conder-Nedela classes, there is no Cayley graph, while in two others, every graph is a Cayley graph. In eight of the remaining ten classes, we give necessary conditions on the order of the graph for it to be Cayley; there is no such condition in the other two. Also we show that in each of those last ten classes, there are infinitely many Cayley graphs and infinitely many non-Cayley graphs.
In 2011, Fang et al. in 9 posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency d, where eitherd≤20or d is a prime number. The only case ...for which the complete solution of this problem is known is of d=3. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valencyd≥4. Even for this problem, it was only solved for the cases when either d≤5 or d=7 and the vertex stabilizer is solvable. In this paper, we make crucial progress towards the above problems by completely solving the second problem for the case when d≥11 is a prime and the vertex stabilizer is solvable.
A graph Γ is G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. If V(Γ) admits a nontrivial G-invariant partition B such that ...for blocks B,C∈B adjacent in the quotient graph ΓB of Γ relative to B, exactly one vertex of B has no neighbour in C, then Γ is called an almost multicover of ΓB. In this case an incidence structure with point set B arises naturally, and it is a (G,2)-point-transitive and G-block-transitive 2-design if in addition ΓB is a complete graph. In this paper we classify all G-symmetric graphs Γ such that (i) B has block size |B|≥3; (ii) ΓB is complete and almost multi-covered by Γ; (iii) the incidence structure involved is a linear space; and (iv) G contains a regular normal subgroup which is elementary abelian. This classification together with earlier results in Gardiner and Praeger (2018), Giulietti et al. (2013) and Fang et al. (2016) completes the classification of symmetric graphs satisfying (i) and (ii).
A 2-geodesic of a graph is a vertex triple (u,v,w) with v adjacent to both u and w, u≠w and u,w are not adjacent. A graph is said to be 2-geodesic-transitive if its automorphism group is transitive ...on both the set of arcs and the set of 2-geodesics. In this paper, we first determine the family of 2-geodesic-transitive graphs which are locally self-complementary, and then classify the family of 2-geodesic-transitive graphs that the local subgraph induced by the neighbor of a vertex is an arc-transitive circulant.
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).
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