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We relate homological properties of a binomial edge ideal JG to invariants that measure the connectivity of a simple graph G. Specifically, we show if R/JG is a Cohen-Macaulay ring, then graph toughness of G is exactly 1/2. We also give an inequality between the depth of R/JG and the vertex-connectivity of G. In addition, we study the Hilbert-Samuel multiplicity and the Hilbert-Kunz multiplicity of R/JG.
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Banerjee et al. (2016) studied this question.
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