Binomial edge ideals and conditional independence statements
TL;DR: It follows that all binomial edge ideals are radical ideals, and the results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones.
About: This article is published in Advances in Applied Mathematics.The article was published on 2010-09-01 and is currently open access. It has received 244 citations till now. The article focuses on the topics: Fractional ideal & Chordal graph.
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TL;DR: In this article, the Grobner basis of I G of S generated by 2-minors [i, j] of X which correspond to edges of G was constructed.
Abstract: Let G be a finite graph on [n] = {1, 2,…, n}, X a 2 × n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this article, we study about ideals I G of S generated by 2-minors [i, j] of X which correspond to edges {i, j} of G. In particular, we construct a Grobner basis of I G as a set of paths of G and compute a primary decomposition.
156 citations
Cites background from "Binomial edge ideals and conditiona..."
...After the first version of this article was written, very recently, it has been brought to my attention that Herzog, Hibi, Hreinsdottir, Kahle, and Rauh wrote an article [ 6 ] which has considerable overlaps with this article....
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TL;DR: In this article, the depth of classes of binomial edge ideals and classifications of closed graphs with Cohen-Macaulay edge ideal were studied and the binomial-edge ideal is defined.
Abstract: We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.
116 citations
TL;DR: In this article, it was shown that the graph AG(R) is connected with diameter at most two and with girth at most four provided that AG(r) has a cycle.
Abstract: Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann R (a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R)* = Z(R)∖{0}, and two distinct vertices x and y are adjacent if and only if ann R (xy) ≠ ann R (x) ∪ ann R (y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). In this article, we study the graph AG(R). For a commutative ring R, we show that AG(R) is connected with diameter at most two and with girth at most four provided that AG(R) has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG(R) is identical to the zero-divisor graph Γ(R) if and only if R has exactly two minimal prime ideals.
108 citations
TL;DR: All graphs whose binomial edge ideals have a linear resolution are characterized, and it is shown that complete graphs are the only graphs with this property.
Abstract: We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of a closed graph.
90 citations
Cites background from "Binomial edge ideals and conditiona..."
...Binomial edge ideals of graphs were introduced in [4]....
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TL;DR: This work characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal I"G.
74 citations
References
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Book•
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01 Jan 1993
TL;DR: In this article, the authors present a self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules.
Abstract: In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.
2,760 citations
Book•
05 Aug 2002
TL;DR: Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science, and it will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.
Abstract: The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.
1,938 citations
TL;DR: An algebra of full binomial type, in the sense of Doubilet-Rota-Stanley, is constructed which yields the generating functions which occur in the above context.
477 citations
Book•
18 Dec 2008
TL;DR: Markov Bases and Likelihood Inference have been used in this article for conditional independence and conditional independence in the context of Bayesian Integrals, and they have been shown to work well with open problems.
Abstract: Markov Bases.- Likelihood Inference.- Conditional Independence.- Hidden Variables.- Bayesian Integrals.- Exercises.- Open Problems.
368 citations
"Binomial edge ideals and conditiona..." refers background in this paper
...Binomial edge ideals, as they are defined in this paper, also arise in the study of conditional independence statements [5]....
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...It is easy to prove this corollary directly using the intersection axiom [5]....
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TL;DR: For a graph G and its associated ideal I(G) as discussed by the authors, a formula for the Krull dimension of the symmetric algebra of G is given along with a description of when this algebra is a domain.
Abstract: For a graph G we consider its associated ideal I(G). We uncover large classes of Cohen-Macaulay (=CM) graphs, in particular the full subclass of CM trees is presented. A formula for the Krull dimension of the symmetric algebra of I(G) is given along with a description of when this algebra is a domain. The first Koszul homology module of a CM tree is also studied.
350 citations
Additional excerpts
...The edge ideal of a graph has been introduced by Villarreal [12] where he studied the Cohen–Macaulay property of such ideals....
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