Gabriel Verret

Cubic vertex-transitive graphs on up to 1280 vertices

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 arc- set, 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.

Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs

The main result of this paper is that, if ? is a connected 4-valent G-arc-transitive graph and v is a vertex of ?, then either ? is part of a well-understood infinite family of graphs, or | G v | ? 2 4 3 6 or 2 | G v | log 2 ? ( | G v | / 2 ) ? | V ? | and that this last bound is tight. As a corollary, we get a similar result for 3-valent vertex-transitive graphs.

A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two

A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 verticesxa0is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a half-arc-transitive group of automorphisms on up to 1000 verticesxa0is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed.

On the order of arc-stabilisers in arc-transitive graphs, II

Let

Cayley graphs on abelian groups

Gamma

On graph-restrictive permutation groups

be a

TETRAVALENT ARC-TRANSITIVE GRAPHS WITH UNBOUNDED VERTEX-STABILIZERS

G

Automorphisms of Cayley graphs on generalised dicyclic groups

-vertex-transitive graph and let

On the vertex-stabiliser in arc-transitive digraphs

(u,v)

Bounding the size of a vertex-stabiliser in a finite vertex-transitive graph

be an arc of