[1] It is the largest distance-transitive graph with degree 11.

[2] The automorphism group of the Livingstone graph is the sporadic simple group J1, and the stabiliser of a point is PSL(2,11).

As the stabiliser is maximal in J1, it acts primitively on the graph.

As the Livingstone graph is distance-transitive, PSL(2,11) acts transitively on the set of 11 vertices adjacent to a reference vertex v, and also on the set of 12 vertices at distance 4 from v. The second action is equivalent to the standard action of PSL(2,11) on the projective line over F11; the first is equivalent to an exceptional action on 11 points, related to the Paley biplane.

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