Livingstone graph

Livingstone graph
Vertices	266
Edges	1463
Radius	4
Diameter	4
Girth	5
Automorphisms	175560 (J1)
Properties	Symmetric Distance-transitive Primitive
Table of graphs and parameters

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}.^[1] It is the largest distance-transitive graph with degree 11 and diameter ≤ 4.^{[citation needed]}

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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