A graph G is called well-covered if all maximal independent sets of
vertices have the same cardinality. A simplicial complex Δ is called
pure if all of its facets have the same cardinality. Let G 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 G with the same well-coveredness
property. Then some necessary and sufficient conditions are presented to
recognize fast when a graph in the class G 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