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 McLaughlin graph is a

strongly regular graph In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have comm ...

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 th ...

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

finite simple group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

, 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 : 27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 = 898,128,000 : ≈ 9. History and properties McL is one of the 26 spor ...

. The automorphism group has rank 3, meaning that its

point stabilizer In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

subgroup divides the remaining 274 vertices into two

orbits In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...

. 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 In the mathematical field of graph theory, the McLaughlin graph is a strongly regular graph with parameters (275,112,30,56), and is the only such graph. The group theorist Jack McLaughlin discovered that the automorphism group of this graph h ...

References

External links

* Individual graphs Regular graphs {{combin-stub