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. In the process, we also construct all tetravalent arc-transitive graphs of order at most 640.2000 Mathematics Subject Classification. 20B25.

Abstract. Let A be an abelian group and let ι be the automorphism of A defined by ι : a → a −1 . A Cayley graph Γ = Cay(A, S) is said to have an automorphism group as small as possible if Aut(Γ) = A ⋊ ι . In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.

A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 vertices is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a 1 2 -arc-transitive group of automorphisms on up to 1000 vertices is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed.

Let Γ be a connected G-vertex-transitive graph, let v be a vertex of Γ and let L = G Γ(v) v be the permutation group induced by the action of the vertex-stabiliser Gv on the neighbourhood Γ(v). Then (Γ, G) is said to be locally-L. A transitive permutation group L is graph-restrictive if there exists a constant c(L) such that, for every locally-L pair (Γ, G) and an arc (u, v) of Γ, the inequality |Guv| ≤ c(L) holds.Using this terminology, the Weiss Conjecture says that primitive groups are graphrestrictive. We propose a very strong generalisation of this conjecture: a group is graphrestrictive if and only if it is semiprimitive. (A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either transitive or semiregular.) Our main result is a proof of one of the two implications of this conjecture, namely that graphrestrictive groups are semiprimitive. We also collect the known results and prove some new ones regarding the other implication.. Then (Γ, G) is 2000 Mathematics Subject Classification. Primary 20B25; Secondary 05E18.