Commuting graphs of boundedly generated semigroups. (English) Zbl 1335.05082
Summary: J. Araújo et al. [Eur. J. Comb. 32, No. 2, 178–197 (2011; Zbl 1227.05161)] pose several problems concerning the construction of arbitrary commuting graphs of semigroups.
We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems.
We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs of semigroups in a wider class (containing the class of nilpotent semigroups), is bounded by the minimal number of generators plus two. We also address a problem concerning knit degree.
We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems.
We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs of semigroups in a wider class (containing the class of nilpotent semigroups), is bounded by the minimal number of generators plus two. We also address a problem concerning knit degree.
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
Citations:
Zbl 1227.05161References:
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| [3] | Araújo, J.; Kinyon, M.; Konieczny, J., Minimal paths in the commuting graphs of semigroups, European J. Combin., 32, 2, 178-197 (2011) · Zbl 1227.05161 |
| [4] | Giudici, M.; Pope, A., The diameters of commuting graphs of linear groups and matrix rings over the integers modulo \(m\), Australas. J. Combin., 48, 221-230 (2010) · Zbl 1232.05114 |
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