Finite 2-Geodesic Transitive Graphs of Prime Valency

Two-geodesic transitive graphs of prime power order

In a non-complete graph $Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $uneq w$ and $u,w$ are not adjacent. The graph $Gamma$ is said to be   $2$-geodesic transitive if its automorphism group is transitive on arcs, and also on 2-geodesics. We first produce a reduction theorem for the family of $2$-geodesic transitive graphs of prime power or...

Primitive Half-transitive Graphs of Valency Twice a Prime

Half-Transitive Group Actions on Finite Graphs of Valency 4

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...

Finite two-distance-transitive graphs of valency 6

A non-complete graph Γ is said to be (G, 2)-distance-transitive if, for i = 1, 2 and for any two vertex pairs (u1, v1) and (u2, v2) with dΓ(u1, v1) = dΓ(u2, v2) = i, there exists g ∈ G such that (u1, v1) = (u2, v2). This paper classifies the family of (G, 2)-distancetransitive graphs of valency 6 which are not (G, 2)-arc-transitive.

FINITE s-ARC TRANSITIVE GRAPHS OF PRIME-POWER ORDER

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...