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Graph connectivity and binomial edge ideals


Authors: Arindam Banerjee and Luis Núñez-Betancourt
Journal: Proc. Amer. Math. Soc. 145 (2017), 487-499
MSC (2010): Primary 13C14, 05C40, 05E40
DOI: https://doi.org/10.1090/proc/13241
Published electronically: August 18, 2016
MathSciNet review: 3577855
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Abstract: We relate homological properties of a binomial edge ideal $ \mathcal {J}_G$ to invariants that measure the connectivity of a simple graph $ G$. Specifically, we show if $ R/\mathcal {J}_G$ is a Cohen-Macaulay ring, then graph toughness of $ G$ is exactly $ \frac {1}{2}$. We also give an inequality between the depth of $ R/\mathcal {J}_G$ and the vertex-connectivity of $ G$. In addition, we study the Hilbert-Samuel multiplicity and the Hilbert-Kunz multiplicity of $ R/\mathcal {J}_G$.


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Additional Information

Arindam Banerjee
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
Email: banerj19@purdue.edu

Luis Núñez-Betancourt
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904-4135 – and – Centro de Investigación en Matemáticas, Guanajuato, Gto., México
Email: luisnub@cimat.mx

DOI: https://doi.org/10.1090/proc/13241
Keywords: Binomial edge ideals, graph toughness, vertex connectivity, Cohen-Macaulay rings
Received by editor(s): January 23, 2016
Received by editor(s) in revised form: April 4, 2016
Published electronically: August 18, 2016
Additional Notes: The second author gratefully acknowledges the support of the National Science Foundation for support through Grant DMS-1502282
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society