Rahim Zaare-Nahandi
Sequentially $S_r$ simplicial complexes and sequentially $S_2$ graphs.
Wed, 04/21/2010 - 15:01 — SomNambulAuthors: Naoki Terai, Siamak Yassemi, Hassan Haghighi, Rahim Zaare-Nahandi
Subjects: Commutative Algebra
AbstractWe introduce sequentially $S_r$ modules over a commutative graded ring and
sequentially $S_r$ simplicial complexes. This generalizes two properties for
modules and simplicial complexes: being sequentially Cohen-Macaulay, and
satisfying Serre's condition $S_r$. In analogy with the sequentially
Cohen-Macaulay property, we show that a simplicial complex is sequentially
$S_r$ if and only if its pure $i$-skeleton is $S_r$ for all $i$. For $r=2$, we
provide a more relaxed characterization.Bipartite $S_2$ graphs are Cohen-Macaulay.
Fri, 01/22/2010 - 16:01 — SomNambulAuthors: Siamak Yassemi, Hassan Haghighi, Rahim Zaare-Nahandi
Subjects: Commutative Algebra
AbstractIn this paper we show that if the Stanley-Reisner ring of the simplicial
complex of independent sets of a bipartite graph $G$ satisfies Serre's
condition $S_2$, then $G$ is Cohen-Macaulay. As a consequence, the
characterization of Cohen-Macaulay bipartite graphs due to Herzog and Hibi
carries over this family of bipartite graphs. We check that the equivalence of
Cohen-Macaulay property and the condition $S_2$ is also true for chordal graphs
and we classify cyclic graphs with respect to the condition $S_2$.
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