Abstract
A graph G is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complexis 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 t o recognize which graphs in the class G are well-covered. This characterization has a nice algebraic interpretation according to zero-divisor elements of edge ring of graphs which is shown in this paper.
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