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 M 11 . 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 M 11 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 M 11 graphs occur as the second and third members of a tower of graphs defined on a conjugacy class of involutions of the simple groups A 5 , PSL ( 2 , 11 ) , M 11 and M 12 with two involutions adjacent if they generate a special S 3 . 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.