Intersection Graphs of $$S$$ S -Acts

Intersection graphs associated with semigroup acts

“…Some papers devoted to intersection graphs derived from other structures such as groups, rings, modules, and lattices have appeared during the years. Inspired by these studies, Rasouli and Tehranian [18] extended the idea to acts over semigroups and verified some elementary properties of the intersection graph Int(A) of non-trivial subacts of an Sact A over a semigroup S. Here we investigate more aspects of this graph and obtain more results. We try to relate the algebraic properties of an S-act to the graph-theoretic properties of its intersection graph.…”

“…It is of interest to know whether a graph is an intersection graph of an Sact. This property holds for every complete graph (see [18,Proposition 2.2]). Here the two classes of bipartite and wheel graphs which are intersection graphs of some S-acts are fully characterized.…”

“…Let A be an S-act. Recall from [18] that the intersection graph of A, denoted by Int(A), is an undirected simple graph whose vertices are non-trivial subacts of A and two distinct vertices B and C are adjacent if and only if B ∩ C = ∅. Here we first investigate some properties of the graph Int(A) for an S-act A. Hereby, the interplay between some algebraic properties of an S-act A and graph-theoretic properties of Int(A) is considered.…”

On the inclusion graphs of S-acts

“…Remark 2.13. In [13], it has shown that the independence number of the intersection graph of an S-act A equals the number of minimal subacts of A. But this is not the case for the inclusion graphs.…”

“…Recently, inclusion graphs attached to rings, vector spaces and groups have been studied in [1,8,5]. Moreover, some works associating graphs to S-acts can be found in [6,9,13].…”