Some graphs related to the small Mathieu groups

Two different constructions are given of a rank 8 arc-transitive graph with 165 vertices and valency 8, whose automorphism group is M11M11. One involves 3-subsets of an 11-set while the other involves 4-subsets of a 12-set, and the constructions are linked with the Witt 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 M11M11 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 M11M11 graphs occur as the second and third members of a tower of graphs defined on a conjugacy class of involutions of the simple groups A5A5, PSL(2,11), M11M11 and M12M12 with two involutions adjacent if they generate a special S3S3. The first graph in the tower is the line graph of the Petersen graph while the fourth is the Johnson graph J(12,4)J(12,4). The graphs also arise as collineation graphs of rank 2 truncations of various residues of certain PP-geometries.