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# Livingstone graph

In the mathematical field of graph theory, the Livingstone graph is a distance-transitive graph with 266 vertices and 1463 edges. It is the largest distance-transitive graph with degree 11.[1]

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.

References

Weisstein, Eric W., "Livingstone Graph", MathWorld.

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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