On the connectivity and equality of some graphs on finite semigroups. (English) Zbl 1510.05104
In this article, four families of graphs associated with semigroups are discussed, which are: the power graphs, the cyclic graphs, the enhanced power graphs and the commuting graphs. In the first part of the article, only semigroups having one idempotent are considered, and in fact, the interconnection between the diameters of the graphs associated with such semigroups is investigated. Consequently, the results on the connectedness and the diameter of properly enhanced power graphs (or cyclic graphs) of finite groups, especially symmetric groups and alternating groups, are obtained. In the second part of this article, for an arbitrary pair of these four families, the finite semigroups are classified such that the related graphs coincide. The obtained results generalize some of the corresponding results of these graphs on groups to semigroups.
Reviewer: Ali Reza Moghaddamfar (Tehran)
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
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