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.

Let Γ be a finite X-symmetric graph with a nontrivial Xinvariant partition B on V (Γ) such that ΓB is a connected (X, 2)-arctransitive graph and Γ is not a multicover of ΓB. A characterization of (Γ,X,B) was given in [20] for the case where |Γ(C) ∩ B| = 2 for B ∈ B and C ∈ ΓB(B). This motivates us to investigate the case where |Γ(C) ∩ B| = 3, that is, Γ[B,C] is isomorphic to one of 3K2, K3,3 − ...

A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is calle...

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 near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ , then the graph is called polygonal. We provide a construction of an infinite family of polygonal graphs of arbitrary even girth with 2-arc transitive automorphism groups, showing that there are infinitely many 2...

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let p and q be primes. It is known that no tetravalent half-arc-transitive graphs of order 2p2 exist and a tetravalent half-arctransitive graph of order 4p must be non-Cayley; such a non-Cayley graph exists if and only if p − 1 is divisible by 8 and it is unique for a given o...

A regular covering projection is called dihedral or abelian if the covering transformation group is dihedral or abelian. A lot of work has been done with regard to the classification of arc-transitive abelian (or elementary abelian, or cyclic) covers of symmetric graphs. In this paper, we investigate arc-transitive dihedral regular covers of symmetric (arc-transitive) cubic graphs. In particula...

An s-arc in a graph is a vertex sequence (α0, α1, . . . , αs) such that {αi−1, αi} ∈ EΓ for 1 6 i 6 s and αi−1 6= αi+1 for 1 6 i 6 s− 1. This paper gives a characterization of a class of s-transitive graphs; that is, graphs for which the automorphism group is transitive on s-arcs but not on (s+ 1)-arcs. It is proved that if Γ is a finite connected s-transitive graph (where s > 2) of order a p-p...

Abstract: When restricted to the rank 1 and rank 2 faces, the Hasse diagram of a regular abstract 4-polytope provides a bipartite graph with a high degree of symmetry. Focusing on the case of the self-dual polytopes of type 3,q,3, I will show that the graphs obtained are 3-arc transitive cubic graphs. Also, given any 3-arc transitive cubic graph, I will discuss when it is possible to consider t...