<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>JG</italic> denote the binomial edge ideal of a connected undirected graph on <italic>n</italic> vertices. This is the ideal generated by the binomials <italic>xiyj − xjyi,</italic> 1 ≤ <italic>i &lt; j≤ n,</italic> in the polynomial ring <italic>S</italic> = <italic>K</italic>[<italic>x</italic>1, . . . , <italic>xn, y</italic>1, . . . , <italic>yn</italic>] where {<italic>i, j</italic>} is an edge of <italic>G</italic>. We study the arithmetic properties of <italic>S/JG</italic> for <italic>G</italic>, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of <italic>S/JG</italic>.</abstract>

Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj − xjyi, 1 ≤ i < j≤ n, in the polynomial ring S = K[x1, . . . , xn, y1, . . . , yn] where {i, j} is an edge of G. We study the arithmetic properties of S/JG for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of S/JG.

Algebraic properties of the binomial edge ideal of a complete bipartite graph

Martin-Luther-Universität Halle-Wittenberg, Institut für Informatik, D — 06 099 Halle (Saale), Germany

Department of Mathematics, University of Management and Technology, Lahore, Pakistan

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Algebraic properties of the binomial edge ideal of a complete bipartite graph"}

Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj − xjyi, 1 ≤ i < j≤ n, in...

Algebraic properties of the binomial edge ideal of a complete bipartite graph

Publié en ligne: 20 oct. 2015

Pages: 217 - 238

Reçu: 01 févr. 2013

Accepté: 01 juin 2013

DOI: https://doi.org/10.2478/auom-2014-0043

Mots clés
binomial edge ideal, complete bipartite graph, pure resolution

© 2014 Peter Schenzel et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Algebraic properties of the binomial edge ideal of a complete bipartite graph

Publié en ligne: 20 oct. 2015

Pages: 217 - 238

Reçu: 01 févr. 2013

Accepté: 01 juin 2013

DOI: https://doi.org/10.2478/auom-2014-0043

Mots clésbinomial edge ideal, complete bipartite graph, pure resolution

© 2014 Peter Schenzel et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
binomial edge ideal, complete bipartite graph, pure resolution