Commuting graph
In mathematics, the commuting graph of a semigroup, or in particular of a group, is an undirected graph in which the vertices are elements of the semigroup and there is an edge between any pair of elements that commute. Commuting graphs have been used to study groups and semigroups by seeking relationships between the combinatorial structure of the graph and the algebraic structure of the group or semigroup.
Depending on the author, the vertex set may comprise every element of the semigroup, or only the non-central elements. If the central elements are excluded, the commuting graph is usually only defined for non-abelian groups and non-commutative semigroups.
For the purposes of this article, the vertices of the commuting graph are the non-central elements unless otherwise noted.
History
The concept of a commuting graph was first introduced for groups in 1955, although the term 'commuting graph' was not coined until 1983. They played a implicit role in Bernd Fischer's discovery of the sporadic groups now known as the Fischer groups.The study of the commuting graphs of semigroups other than groups was initiated in 2011.
Properties
Connectedness and diameters
It is possible for a commuting graph to be non-connected and thus not to have a finite diameter.For a finite set, the commuting graph of the symmetric group is connected if and only if and are non-prime, and the commuting graph of the alternating group is connected if and only if,, and are non-prime. When connected, the commuting graphs of and have diameter at most 5.
The commuting graph of the symmetric inverse semigroup is not connected if and only if is an odd prime. When is not an odd prime, it has diameter 4 or 5, and is known to have diameter 4 when is even and diameter 5 when is a power of an odd prime.
For every natural number, there is a finite group whose commuting graph is connected and has diameter equal to. But if a finite group has trivial center and its commuting graph is connected, then its diameter is at most 10.
The commuting graph of a completely simple semigroup is never connected except when it is a group, and if it not a group, its connected components are the commuting graphs including central elements of its maximal subgroups.