Equality of Power Graphs, Enhanced Power Graphs and Commuting Graphs on Semigroups

This paper focuses on three kinds of graphs built from semigroups: power graphs, enhanced power graphs and commuting graphs. Let S be a semigroup constructed in a prescribed way from a semigroup T. We show in particular instances that pairs of these graphs built over S coincide precisely when the same pairs built over T coincide. This is true, for example, when S is a completely simple semigroup and T is a maximal subgroup. We give several other instances of this phenomenon. In addition, we give counterexamples to show that equality for an arbitrary pair of these three graphs does not always pass between semigroups and their maximal subgroups.