An equivalent relation between transitive 1-factorizations of arctransitive graphs and factorizations of their automorphism groups is established. This relation provides a method for constructing and characterizing transitive 1-factorizations for certain classes of arc-transitive graphs. Then a characterization is given of 2-arc-transitive graphs which have transitive 1factorizations. In this c...

We prove that the underlying graphs of p-gonal r-valent orientably regular maps are arc-transitive but non-Cayley if r ≥ 3 and p is a prime greater than r(r − 1).

The generalised Paley graphs, as the name implies, are a generalisation of the Paley graphs which are well-known to be self-complementary and arc-transitive. They are defined as follows. Let F be a finite field of order p, where p is a prime, and let S be a subgroup of the multiplicative group of F, where S must have even order if p is odd. The generalised Paley graph of F relative to S has ver...

Let Γ be a G-symmetric graph, and let B be a nontrivial G-invariant partition of the vertex set of Γ . This paper aims to characterize (Γ ,G) under the conditions that the quotient graph ΓB is (G, 2)-arc transitive and the induced subgraph between two adjacent blocks is 2 · K2 or K2,2. The results answer two questions about the relationship between Γ and ΓB for this class of graphs. c © 2007 El...

A 1 2 -arc-transitive graph is a vertexand edgebut not arc-transitive graph. In all known constructions of quartic 1 2 -arc-transitive graphs, vertex stabilizers are isomorphic to Z 2,Z 2 2 or to D8. In this article, for each positive integer m ≥ 1, an infinite family of quartic 1 2 -arctransitive graphs having vertex stabilizers isomorphic to Z m

The action of a subgroup G of automorphisms of a graph X is said to be 2 -transitive if it is vertexand edgebut not arc-transitive. In this case the graph X is said to be (G, 2)-transitive. In particular, X is 1 2 -transitive if it is (Aut X, 1 2)-transitive. The 2 -transitive action of G on X induces an orientation of the edges of X which is preserved by G. Let X have valency 4. An even length...

We study the class of edge-transitive graphs of square-free order and valency at most k. It is shown that, except for a few special families of graphs, only finitely many members in this class are basic (namely, not a normal multicover of another member). Using this result, we determine the automorphism groups of locally primitive arc-transitive graphs with square-free order.

Let be a G-symmetric graph admitting a nontrivial G-invariant partition B. For B∈B, let D(B)= (B; B(B); I) be the 1-design in which IC for ∈B and C ∈ B(B) if and only if is adjacent to at least one vertex of C, where B(B) is the neighbourhood of B in the quotient graph B of relative to B. In a natural way the setwise stabilizer GB of B in G induces a group of automorphisms of D(B). In this pape...

Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space of the form ProjSK(V ), where V be a k-central two-sided vector space over K of rank two and SK(V ) is the noncommutative symmetric algebra generated by V over K defined by M. Van den Bergh [26]. We study the geometry of these spaces. More precisely, we pr...

In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k 6= 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs...