A note on Non-inclusion principal ideal graph of Completely simple semigroups

Résumé

The non-inclusion principal left ideal graph of a semigroup,
denoted by $\mathbf{nPiG}_{l}(S)$ is a simple, undirected graph with the nonzero elements of $S$
as vertices and two distinct elements $a, b \in S$ are adjacent if and only if $a \notin S^1b$
and $b\notin S^1a$, where $S^1a$ and $S^1b$ are principal left ideals generated by $a$ and $b$ respectively.
The non-inclusion principal right ideal graph, $\mathbf{nPiG}_{r}(S)$ is defined similarly. Here, we identify
the structure of $\mathbf{nPiG}_{l}(S)$ and $\mathbf{nPiG}_{r}(S)$ when $S$ is a completely simple semigroup
in terms of Green's equivalences and we establish the correspondence between these graphs and the complete
$k$-partite graphs. Furthermore, we analyze the automorphism groups and discuss some energies associated
with these graph structures.