Intersection graphs of \(S\)-acts
The intersection graph \(G(A)\) of an \(S\)-act \(A\) is undirected, has the non-trivial subacts of \(A\) as vertices, two of which are adjacent if their intersection is non-empty. Let \(S\) be a monoid. The authors prove for example: If \(G(A)\) is connected then \(\mathrm{diam}(G(A))\leq 2\), if it contains a cycle then \(\mathrm{girth}(G(A))=3\). Suppose that \(A\) is non-cyclic. If \(A\) is free or projective then \(G(A)\) is connected iff \(|G(A)|\geq 6\), in this case \(\mathrm{diam}(G(A))= 2\) and \(\mathrm{girth}(G(A))=3\). Similar results are presented if \(S\) is a group.
scientific article; zbMATH DE number 6494660