An infinite Family of 4‐Arc‐Transitive Cubic Graphs Each with Girth 12

If p is any prime, and θ is that automorphism of the group SL(3, p ) which takes each matrix to the transpose of its inverse, then there exists a connected trivalent graph Γ(p) on1 12p 3 ( p 3 − 1 ) ( p 2 − 1 ) vertices with the split extension SL(3, p )〈θ〉 as a group of automorphisms acting regularly on its 4‐arcs. In fact if p ≠ 3 then this group is the full automorphism group of Γ( p ), while the graph Γ(3) is 5‐arc‐transitive with full automorphism group SL(3,3)〈0〉 × C 2 . The girth of Γ( p ) is 12, except in th case p = 2 (where the girth is 6). Furthermore, in all cases Γ( p ) is bipartite, with SL(3, p ) fixing each part. Also when p ≡ 1 mod 3 the graph Γ( p ) is a triple cover of another trivalent graph, which has automorphism group PSL(3, p )〈0〉 acting regularly on its 4‐arcs. These claims are proved using elementary theory of symmetric graphs, together with a suitable choice of three matrices which generate SL(3, Z ). They also provide a proof that the group 4 + ( a 12 ) described by Biggs in Computational group theor (ed. M. Atkinson) is infinite.