TY - JOUR
AU - Conder, Marston
AB - 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) on 112p3 (p3−1)(p2−1) 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〉 × C2. 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+(a12) described by Biggs in Computational group theor(ed. M. Atkinson) is infinite.
TI - An infinite Family of 4‐Arc‐Transitive Cubic Graphs Each with Girth 12
JF - Bulletin of the London Mathematical Society
DO - 10.1112/blms/21.4.375
DA - 1989-07-01
UR - https://www.deepdyve.com/lp/wiley/an-infinite-family-of-4-arc-transitive-cubic-graphs-each-with-girth-12-R1LCRmMoRF
SP - 375
EP - 380
VL - 21
IS - 4
DP - DeepDyve
ER -