33 research outputs found
Pure simplicial complexes and well-covered graphs
A graph is called well-covered if all maximal independent sets of
vertices have the same cardinality. A simplicial complex is called
pure if all of its facets have the same cardinality. Let be the
class of graphs with some disjoint maximal cliques covering all vertices. In
this paper, we prove that for any simplicial complex or any graph, there is a
corresponding graph in class with the same well-coveredness
property. Then some necessary and sufficient conditions are presented to
recognize fast when a graph in the class is well-covered or not. To do
this characterization, we use an algebraic interpretation according to
zero-divisor elements of the edge rings of graphs.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1009.524
Cohen-Macaulayness of bipartite graphs, revisited
Cohen-Macaulayness of bipartite graphs is investigated by several
mathematicians and has been characterized combinatorially. In this note, we
give some different combinatorial conditions for a bipartite graph which are
equal to Cohen-Macaulayness of the graphs. Conditions in the previous works are
depending on an appropriate ordering on vertices of the graph. The conditions
presented in this paper are not depending to any ordering. Finally, we present
a fast algorithm to check Cohen-Macaulayness of a given bipartite graph.Comment: 7 page
Aluffi torsion-free ideals
A special class of algebras which are intermediate between the symmetric and
the Rees algebras of an ideal was introduced by P. Aluffi in 2004 to define
characteristic cycle of a hypersurface parallel to conormal cycle in
intersection theory. These algebras are recently investigated by A. Nasrollah
Nejad and A. Simis who named them Aluffi algebras. For a pair of ideals
of a commutative ring , the Aluffi algebra of is called
Aluffi torsion-free if it is isomorphic to the Rees algebra of . In this
paper, ideals generated by 2-minors of a matrix of linear forms and
also edge ideals of graphs are considered and some conditions are presented
which are equivalent to Aluffi torsion-free property of them. Also many other
examples and further questions are presented.Comment: 28 page