First and foremost, I have extensive experience with permutation group theory. There is an extensive literature on groups that act primitively on a set Ω, i.e., those groups which do not preserve a nontrivial system of blocks. Furthering this is the notion of quasiprimitivity. A group G acts quasiprimitively on a set Ω if every nontrivial normal subgroup of G is transitive on Ω. Every group act...

Finite graphs of valency 4 and girth 4 admitting 1 2-transitive group actions, that is vertex-and edge-but not arc-transitive group actions, are investigated. A graph is said to be 1 2-transitive if its automorphism group acts 1 2-transitively. There is a natural (balanced) orientation of the edge set of a 1 2-transitive graph induced and preserved by its auto-morphism group. Among others it is...

Characterizing regular covers of edge-transitive or arc-transitive graphs is currently a hot topic in algebraic graph theory. In this paper, we will classify arctransitive regular cyclic covers of the complete bipartite graph Kp,p for each odd prime p. The classification consists of four infinite families of graphs. In particular, such covers exist for each odd prime p. The regular elementary a...

We study (G, 2)-arc-transitive graphs for innately transitive permutation groups G such that G can be embedded into a wreath product SymΓwr Sl acting in product action on Γ. We find two such connected graphs: the first is Sylvester’s double six graph with 36 vertices, while the second is a graph with 120 vertices whose automorphism group is Aut Sp(4, 4). We prove that under certain conditions n...

Let ? be a finite, undirected, connected, simple graph. We say that matching M is permutable m -matching if contains edges and the subgroup of Aut ( ) fixes setwise allows to permuted in any fashion. A 2-transitive stabilizer can map ordered pair distinct other . provide constructions graphs with matching; we show that, an arc-transitive graph for ? 4 , then degree at least ; and, when sufficie...

One version of the polycirculant conjecture states that every vertex-transitive graph has a non-identity semiregular automorphism that is, a non-identity automorphism whose cycles all have the same length. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open valency.