Pure simplicial complexes and well-covered graphs

Details Pure simplicial complexes and well-covered graphs Pure simplicial complexes and well-covered graphs

2011-04-23

arxiv.org/abs/1104.4556v1

Mathematics

Commutative Algebra

9 pages

Scientific paper

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complex $\Delta$ is called pure if all of its facets have the same cardinality. Let $\mathcal G$ 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 $\mathcal G$ with the same well-coveredness property. Then some necessary and sufficient conditions are presented to recognize which graphs in the class $\cal G$ 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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