Pure simplicial complexes and well-covered graphs

TL;DR

This paper explores the relationship between pure simplicial complexes and well-covered graphs, providing conditions to identify well-covered graphs within a specific class using algebraic methods.

Contribution

It establishes a correspondence between simplicial complexes and graphs in class G with the same well-coveredness, and offers criteria for recognizing well-covered graphs algebraically.

Findings

Graphs in class G can be characterized by algebraic properties of their edge rings.

Necessary and sufficient conditions for recognizing well-covered graphs are provided.

A correspondence between pure simplicial complexes and well-covered graphs is established.

Abstract

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 fast when a graph in the class $\cal G$ 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.