Non-absorbing product graph of completely simple semigroups

Abstract

We introduce the non-absorbing product graph of a semigroup S, an undirected graph whose vertices are the elements of S and where two distinct vertices are adjacent precisely when their product differs from both factors. Focusing on completely simple semigroups, we establish a structural decomposition of this graph as the edge-union of a complete graph on the non-idempotent elements, a regular bipartite graph connecting idempotents to non-idempotents, and a regular multipartite graph on the idempotents.   Moreover, we analyze the absorbing product graph, the complement of the non-absorbing product graph, which decomposes into a regular bipartite graph together with complete subgraphs arising from Green's classes of idempotents. Unlike its counterpart, it is always connected yet fails to be Hamiltonian in general. This natural partition of the vertex set into idempotents and non‑idempotents reflects the core algebraic structure in a purely graphical way.