We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc ...transitive or the valency p satisfies p≡1(mod4), and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph Kp+1 with automorphism group PSL(2,p)×Z2 and diameter 3.
In this sequel to the paper ‘Arc-transitive abelian regular covers of cubic graphs’, all arc-transitive abelian regular covers of the Heawood graph are found. These covers include graphs that are ...1-arc-regular, and others that are 4-arc-regular (like the Heawood graph). Remarkably, also some of these covers are 2-arc-regular.
Regular covers of complete graphs whose fibre-preserving automorphism groups act 2-arc-transitively are investigated. Such covers have been classified when the covering transformation groups K are ...cyclic groups Zd for an integer d≥2, metacyclic abelian groups Zp2, or nonmetacyclic abelian groups Zp3 for a prime p (see S.F. Du et al. (1998) 5 for the first two metacyclic group cases and see S.F. Du et al. (2005) 3 for the third nonmetacyclic group case). In this paper, a complete classification is achieved of all such covers when K is any metacyclic group.
A Cayley graph
is said to be
if its full automorphism group Aut
is transitive on the arc set of
. In this paper we give a characterization of pentavalent arc-transitive Cayley graphs on a class of ...Frobenius groups with soluble vertex stabilizer.
A finite graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on $V(\Gamma)$ and transitively on the set of ordered pairs of adjacent vertices of $\Gamma$. ...If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks $B, C \in {\cal B}$ adjacent in the quotient graph $\Gamma_{{\cal B}}$ relative to ${\cal B}$, exactly one vertex of $B$ has no neighbour in $C$, then we say that $\Gamma$ is an almost multicover of $\Gamma_{{\cal B}}$. In this case there arises a natural incidence structure ${\cal D}(\Gamma, {\cal B})$ with point set ${\cal B}$. If in addition $\Gamma_{{\cal B}}$ is a complete graph, then ${\cal D}(\Gamma, {\cal B})$ is a $(G, 2)$-point-transitive and $G$-block-transitive $2$-$(|{\cal B}|, m+1, \lambda)$ design for some $m \geq 1$, and moreover either $\lambda=1$ or $\lambda=m+1$. In this paper we classify such graphs in the case when $\lambda = m+1$; this together with earlier classifications when $\lambda = 1$ gives a complete classification of almost multicovers of complete graphs.
Unitary graphs are arc‐transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent ...classification of a class of arc‐transitive graphs that admit an automorphism group acting imprimitively on the vertices. In this article, we prove that all unitary graphs are connected of diameter two and girth three. Based on this, we obtain, for any prime power q>2, a lower bound of order O(Δ5/3) on the maximum number of vertices in an arc‐transitive graph of degree Δ=q(q2−1) and diameter two.
A graph is
half-arc-transitive
if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let
p
be a prime. It is known that there exist no tetravalent ...half-arc-transitive graphs of order
p
or 2
p
. Feng et al. (J Algebraic Combin 26:431–451,
2007
) gave the classification of tetravalent half-arc-transitive graphs of order 4
p
. In this paper, a classification is given of all tetravalent half-arc-transitive graphs of order 8
p
.
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