Some graphs related to the small Mathieu groups

Two different constructions are given of a rank 8 arc-transitive graphwith 165 vertices and valency 8,whose automorphismgroup isM11. One involves 3-subsets of an 11-set while the other involves 4-subsets of a 12-set, and the constructions are linkedwith theWitt designs on 11, 12 and 24 points. Four different constructions are given of a rank 9 arc-transitive graph with 55 vertices and valency 6 whose automorphism group is PSL(2, 11). This graph occurs as a subgraph of the M11 graph, and two of the constructions involve 2-subsets of an 11-set while the remaining two involve 3-subsets of an 11-set. The PSL(2, 11) and M11 graphs occur as the second and third members of a tower of graphs defined on a conjugacy class of involutions of the simple groups A5, PSL(2, 11), M11 and M12 with two involutions adjacent if they generate a special S3. The first graph in the tower is the line graph of the Petersen graph while the fourth is the Johnson graph J(12, 4). The graphs also arise as collineation graphs of rank 2 truncations of various residues of certain P-geometries. © 2009 Elsevier Ltd. All rights reserved.