(S2) condition Betti number Biomedical engineering Bipartite graphs Castelnuovo-Mumford regularity Castelnuovo–Mumford regularity Cohen Macaulay Cohen-Macaulay Gröbner bases O-sequence Q1-390 QA1-939 Rees algebra Stanley depth Stanley-Reisner ideal Stanley-Reisner ring Stanley’s inequality algebras with straightening laws associated graded rings bic Book Industry Communication::P Mathematics & science bipartite graph chain polytope circuit circulant graphs colon ideals complete intersection cover ideal depth depth of powers of bipartite graphs directed cycle