In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, the McLaughlin graph is a

strongly regular graph In graph theory, a strongly regular graph (SRG) is a regular graph with vertices and degree such that for some given integers \lambda, \mu \ge 0 * every two adjacent vertices have common neighbours, and * every two non-adjacent vertices h ...

with parameters (275, 112, 30, 56) and is the only such graph. The group theorist Jack McLaughlin discovered that 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 the g ...

of this graph had a subgroup of index 2 which was a previously undiscovered

finite simple group In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple g ...

, now called the

McLaughlin sporadic group In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order : 898,128,000 = 27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 : ≈ 9. History and properties McL is one of the 26 sp ...

. The automorphism group has rank 3, meaning that its

point stabilizer In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under funct ...

subgroup divides the remaining 274 vertices into two

orbits In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...

. Those orbits contain 112 and 162 vertices. The former is the colinearity graph of the generalized quadrangle GQ(3,9). The latter is a strongly regular graph called the local McLaughlin graph.

References

External links

* Individual graphs Regular graphs {{graph-stub