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Mathematics 2016, 4(2), 37; doi:10.3390/math4020037

Smoothness in Binomial Edge Ideals

Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave, Tehran 15914, Iran
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Author to whom correspondence should be addressed.
Academic Editor: J. Alberto Conejero
Received: 10 March 2016 / Revised: 7 May 2016 / Accepted: 17 May 2016 / Published: 1 June 2016
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Abstract

In this paper we study some geometric properties of the algebraic set associated to the binomial edge ideal of a graph. We study the singularity and smoothness of the algebraic set associated to the binomial edge ideal of a graph. Some of these algebraic sets are irreducible and some of them are reducible. If every irreducible component of the algebraic set is smooth we call the graph an edge smooth graph, otherwise it is called an edge singular graph. We show that complete graphs are edge smooth and introduce two conditions such that the graph G is edge singular if and only if it satisfies these conditions. Then, it is shown that cycles and most of trees are edge singular. In addition, it is proved that complete bipartite graphs are edge smooth. View Full-Text
Keywords: binomial edge ideal; edge smooth; edge singular binomial edge ideal; edge smooth; edge singular
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Damadi, H.; Rahmati, F. Smoothness in Binomial Edge Ideals. Mathematics 2016, 4, 37.

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