Livingstone graph
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.[2]
| 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 | |
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