We resolve two problems of [Cameron, Praeger, and Wormald – Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each ‘building block’ is a finite bipartite graph that is...

A circulant is a Cayley graph of a cyclic group Arc transitive circulants of square free order are classi ed It is shown that an arc transitive circulant of square free order n is one of the following the lexicographic product Kb or the deleted lexicographic Kb b where n bm and is an arc transitive circulant or is a normal circulant that is Aut has a normal regular cyclic subgroup

We explore the relation between vertexand edge-transitivity and arc-transitivity of various graphs. We exhibit several families of graphs whose vertexand edgetransitivity imply arctransitivity. In particular, we show that any vertexand edgetransitive graph with twice a prime number of vertices is arctransitive by simplifying the proof of a theorem by Cheng and Oxley, in which they classify all ...

For a positive integer s, a graph Γ is called s-arc transitive if its full automorphism group AutΓ acts transitively on the set of s-arcs of Γ . Given a group G and a subset S of G with S = S−1 and 1 / ∈ S, let Γ = Cay(G, S) be the Cayley graph of G with respect to S and G R the set of right translations of G on G. Then G R forms a regular subgroup of AutΓ . A Cayley graph Γ = Cay(G, S) is call...

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. It was shown by [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent halftransitive graphs of order 4p, European J. Combin. 28 (2007) 726–733] that all tetravalent half-arc-transitive graphs of order 4p for a prime p are non-Cayley and such graphs exist if andonly if p−1 is divi...

A graph is called cubic and tetravalent if all of its vertices have valency 3 and 4, respectively. It is called vertex-transitive and arc-transitive if its automorphism group acts transitively on its vertex-set and on its arcset, respectively. In this paper, we combine some new theoretical results with computer calculations to construct all cubic vertex-transitive graphs of order at most 1280. ...

A graph is said to be 1 2-transitive if its automorphism group is ver-tex and edge but not arc-transitive. For each n 11, a 1 2-transitive graph of valency 4 and girth 6, with the automorphism group isomor-phic to A n Z 2 , is given.

Let 0 be a simple graph and let G be a group of automorphisms of 0. The graph is (G, 2)-arc transitive if G is transitive on the set of the 2-arcs of 0. In this paper we construct a new family of (PSU(3, q2), 2)-arc transitive graphs 0 of valency 9 such that Aut0 = Z3.G, for some almost simple group G with socle PSU(3, q2). This gives a new infinite family of non-quasiprimitive almost simple gr...

We study locally s-arc-transitive graphs arising from the quasiprimitive product action (PA). prove that, for any (G,2)-arc-transitive graph with G acting quasiprimitively type PA on both G-orbits of vertices, group does not act primitively either orbit. Moreover, we construct first examples that are standard double covers type, answering existence question these graphs.

It is known that there are precisely three transitive permutation groups of degree 6 admit an invariant partition with parts size 2 such the kernel action on has order 4; these called A 4 ( ) , S d and c . For each L ∈ { } we construct infinite family finite connected 6-valent graphs Γ n N arc-transitive G ≤ Aut group induced by vertex-stabiliser v neighbourhood a vertex isomorphic to L, ∣ expo...