The Non-Nilpotent Graph of a Semigroup

We associate a graph ${\mathcal N}_{S}$ with a semigroup $S$ (called the upper non-nilpotent graph of $S$). The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). In case $S$ is a group this graph has been introduced by A. Abdollahi and M. Zarrin and some remarkable properties have been proved. The aim of this paper is to study this graph (and some related graphs, such as the non-commuting graph) and to discover the algebraic structure of $S$ determined by the associated graph. It is shown that if a finite semigroup $S$ has empty upper non-nilpotent graph then $S$ is positively Engel. On the other hand, a semigroup has a complete upper non-nilpotent graph if and only if it is a completely simple semigroup that is a band. One of the main results states that if all connected ${\mathcal N}_{S}$-components of a semigroup $S$ are complete (with at least two elements) then $S$ is a band that is a semilattice of its connected components and, moreover, $S$ is an iterated total ideal extension of its connected components. We also show that some graphs, such as a cycle $C_{n}$ on $n$ vertices (with $n\geq 5$), are not the upper non-nilpotent graph of a semigroup. Also, there is precisely one graph on 4 vertices that is not the upper non-nilpotent graph of a semigroup with 4 elements. This work also is a continuation of earlier work by Okni\'nski, Riley and the first named author on (Malcev) nilpotent semigroups.