A remarkable Mathieu graph tower∗

Two different constructions are given of a rank 8 arc-transitive graph with 165 vertices and valency 8, whose automorphism group is M11. 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 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 first author is a Postdoctoral Researcher of the Fonds National de la Recherche Scientifique (Belgium). The second author holds an Australian Postdoctoral Fellowship while the third author holds a QEII Fellowship. A major part of the research of this paper was done during a visit to UWA by the first author which was supported by an ARC Discovery grant of the remaining three authors and by the FNRS.