In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, the Livingstone graph is a

distance-transitive graph In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices and at any distance , and any other two vertices and at the same distance, there is an automorphism of the graph that carri ...

with 266 vertices and 1463 edges. Its

intersection array In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and . ...

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

Algebraic properties

The

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is th ...

of the Livingstone graph is the

sporadic In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. Th ...

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The da ...

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 In combinatorics, combinatorial mathematics, a block design is an incidence structure consisting of a set together with a Family of sets, family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain condition ...

References

Individual graphs Regular graphs {{combin-stub