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 (that is, there is an edge between vertices x and y if and only if xy=yx in the semigroup). 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 (since the central elements — those elements of a semigroup that commute with every other element — would always form a complete subgraph, every vertex of which would be adjacent to every vertex of the whole commuting graph). If the central elements are excluded, the commuting graph is usually only defined for non-abelian groups and non-commutative semigroups (since in these cases the commuting graph would be empty).
For the purposes of this article, the vertices of the commuting graph are the non-central elements unless otherwise noted.
History
[edit]The concept of a commuting graph was first introduced for groups in 1955,[1] although the term 'commuting graph' was not coined until 1983.[2] They played a implicit role in Bernd Fischer's discovery of the sporadic groups now known as the Fischer groups.[3]
The study of the commuting graphs of semigroups other than groups was initiated in 2011.[4]
Properties
[edit]Connectedness and diameters
[edit]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.[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.[6]
For every natural number n, there is a finite group whose commuting graph is connected and has diameter equal to n.[7] But if a finite group has trivial center and its commuting graph is connected, then its diameter is at most 10.[8]
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[9] (which, by the Rees–Suschkewitsch theorem, are isomorphic[10]).
Simple groups
[edit]Non-abelian finite simple groups are uniquely characterized by their commuting graphs, in the sense that if G is a non-abelian finite simple group and H is a group, and the commuting graphs of G and the commuting graph of H are isomorphic (as graphs), then G and H are isomorphic (as groups). This result was conjectured in 2006[11] and proved by different authors for sporadic groups,[12], alternating groups,[13] and groups of Lie type.[14]
Notes
[edit]- ^ Brauer, Richard; Fowler, K. A. (November 1955). "On Groups of Even Order". The Annals of Mathematics. 62 (3): 565. doi:10.2307/1970080.
- ^ Bertram, Edward A. (1983). "Some applications of graph theory to finite groups". Discrete Mathematics. 44 (1): 31–43. doi:10.1016/0012-365X(83)90004-3.
- ^ Fischer, Bernd (September 1971). "Finite groups generated by 3-transpositions. I". Inventiones Mathematicae. 13 (3): 232–246. doi:10.1007/BF01404633.
- ^ Araújo, João; Kinyon, Michael; Konieczny, Janusz (February 2011). "Minimal paths in the commuting graphs of semigroups". European Journal of Combinatorics. 32 (2): 178–197. doi:10.1016/j.ejc.2010.09.004.
- ^ Iranmanesh, A.; Jafarzadeh, A. (February 2008). "On the commuting graph associtated with the symmetric and alternating groups". Journal of Algebra and Its Applications. 07 (01): 129–146. doi:10.1142/S0219498808002710.
- ^ Araújo, João; Bentz, Wolfram; Janusz, Konieczny (April 2015). "The commuting graph of the symmetric inverse semigroup". Israel Journal of Mathematics. 207 (1): 103–149. doi:10.1007/s11856-015-1173-9.
- ^ Cutolo, Giovanni (1 November 2022). "On a construction by Giudici and Parker on commuting graphs of groups". Journal of Combinatorial Theory, Series A. 192 105666. doi:10.1016/j.jcta.2022.105666.
- ^ Morgan, G.L.; Parker, C.W. (November 2013). "The diameter of the commuting graph of a finite group with trivial centre". Journal of Algebra. 393: 41–59. doi:10.1016/j.jalgebra.2013.06.031.
- ^ Paulista, Tânia (3 October 2025). "Commuting graphs of completely simple semigroups". Communications in Algebra. 53 (10): 4215–4226. doi:10.1080/00927872.2025.2481079.
- ^ Howie, John M. (1995). Fundamentals of Semigroup Theory. Oxford: Clarendon Press. p. 77. ISBN 0-19-851194-9.
- ^ Abdollahi, A.; Akbari, S.; Maimani, H.R. (April 2006). "Non-commuting graph of a group". Journal of Algebra. 298 (2): 468–492. doi:10.1016/j.jalgebra.2006.02.015.
- ^ Han, Zhangjia; Chen, Guiyun; Guo, Xiuyun (November 2008). "A characterization theorem for sporadic simple groups". Siberian Mathematical Journal. 49 (6): 1138–1146. doi:10.1007/s11202-008-0111-z.
- ^ Abdollahi, Alireza; Shahverdi, Hamid (May 2012). "Characterization of the alternating group by its non-commuting graph". Journal of Algebra. 357: 203–207. doi:10.1016/j.jalgebra.2012.01.038.
- ^ Solomon, Ronald M.; Woldar, Andrew J. (1 November 2013). "Simple groups are characterized by their non-commuting graphs". Journal of Group Theory. 16 (6): 793–824. doi:10.1515/jgt-2013-0021.