Livingstone graph
{{infobox graph
| name = Livingstone graph
| image = Livingstone graph.svg
| image_caption =
| namesake =
| vertices = 266
| edges = 1463
| automorphisms= 175560 (J1)
| girth = 5
| radius = 4
| diameter = 4
| chromatic_number =
| chromatic_index =
| properties = Symmetric
Distance-transitive
Primitive
}}
In the mathematical field of graph theory, the Livingstone graph is a distance-transitive graph with 266 vertices and 1463 edges. Its intersection array is {11,10,6,1;1,1,5,11}.[https://www.math.mun.ca/distanceregular/graphs/livingstone.html distanceregular.org page on Livingstone Graph] It is the largest distance-transitive graph with degree 11.{{MathWorld|urlname=LivingstoneGraph|title=Livingstone Graph}}
Algebraic properties
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.