Binomial edge ideals and conditional independence statements
We introduce binomial edge ideals attached to a simple graph G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced sq...
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| Veröffentlicht in: | Advances in applied mathematics 2010-09, Vol.45 (3), p.317-333 |
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| container_title | Advances in applied mathematics |
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| creator | Herzog, Jürgen Hibi, Takayuki Hreinsdóttir, Freyja Kahle, Thomas Rauh, Johannes |
| description | We introduce binomial edge ideals attached to a simple graph
G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general
G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of
G. We provide sufficient conditions for Cohen–Macaulayness for closed and nonclosed graphs.
Binomial edge ideals arise naturally in the study of conditional independence ideals. Our 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. In this case the primary decomposition has a natural statistical interpretation. |
| doi_str_mv | 10.1016/j.aam.2010.01.003 |
| format | Article |
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G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general
G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of
G. We provide sufficient conditions for Cohen–Macaulayness for closed and nonclosed graphs.
Binomial edge ideals arise naturally in the study of conditional independence ideals. Our 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. In this case the primary decomposition has a natural statistical interpretation.</description><identifier>ISSN: 0196-8858</identifier><identifier>EISSN: 1090-2074</identifier><identifier>DOI: 10.1016/j.aam.2010.01.003</identifier><identifier>CODEN: AAPMEF</identifier><language>eng</language><publisher>San Diego, CA: Elsevier Inc</publisher><subject>Algebra ; Binomial ideals ; Binomials ; Cohen–Macaulay rings ; Collection ; Commutative rings and algebras ; Conditional independence ideals ; Edge ideals ; Exact sciences and technology ; General mathematics ; General, history and biography ; Generators ; Graphs ; Mathematical analysis ; Mathematical models ; Mathematics ; Number theory ; Numerical analysis ; Numerical analysis. Scientific computation ; Radicals ; Robustness ; Sciences and techniques of general use</subject><ispartof>Advances in applied mathematics, 2010-09, Vol.45 (3), p.317-333</ispartof><rights>2010 Elsevier Inc.</rights><rights>2015 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c468t-38ceaebebf358b9677ff57e2d98c9a41d66ffd0b5a5e114595ab6900258727943</citedby><cites>FETCH-LOGICAL-c468t-38ceaebebf358b9677ff57e2d98c9a41d66ffd0b5a5e114595ab6900258727943</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S019688581000014X$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,3537,27901,27902,65306</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=23146170$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Herzog, Jürgen</creatorcontrib><creatorcontrib>Hibi, Takayuki</creatorcontrib><creatorcontrib>Hreinsdóttir, Freyja</creatorcontrib><creatorcontrib>Kahle, Thomas</creatorcontrib><creatorcontrib>Rauh, Johannes</creatorcontrib><title>Binomial edge ideals and conditional independence statements</title><title>Advances in applied mathematics</title><description>We introduce binomial edge ideals attached to a simple graph
G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general
G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of
G. We provide sufficient conditions for Cohen–Macaulayness for closed and nonclosed graphs.
Binomial edge ideals arise naturally in the study of conditional independence ideals. Our 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. In this case the primary decomposition has a natural statistical interpretation.</description><subject>Algebra</subject><subject>Binomial ideals</subject><subject>Binomials</subject><subject>Cohen–Macaulay rings</subject><subject>Collection</subject><subject>Commutative rings and algebras</subject><subject>Conditional independence ideals</subject><subject>Edge ideals</subject><subject>Exact sciences and technology</subject><subject>General mathematics</subject><subject>General, history and biography</subject><subject>Generators</subject><subject>Graphs</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Number theory</subject><subject>Numerical analysis</subject><subject>Numerical analysis. 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Scientific computation</topic><topic>Radicals</topic><topic>Robustness</topic><topic>Sciences and techniques of general use</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Herzog, Jürgen</creatorcontrib><creatorcontrib>Hibi, Takayuki</creatorcontrib><creatorcontrib>Hreinsdóttir, Freyja</creatorcontrib><creatorcontrib>Kahle, Thomas</creatorcontrib><creatorcontrib>Rauh, Johannes</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Advances in applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Herzog, Jürgen</au><au>Hibi, Takayuki</au><au>Hreinsdóttir, Freyja</au><au>Kahle, Thomas</au><au>Rauh, Johannes</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Binomial edge ideals and conditional independence statements</atitle><jtitle>Advances in applied mathematics</jtitle><date>2010-09-01</date><risdate>2010</risdate><volume>45</volume><issue>3</issue><spage>317</spage><epage>333</epage><pages>317-333</pages><issn>0196-8858</issn><eissn>1090-2074</eissn><coden>AAPMEF</coden><abstract>We introduce binomial edge ideals attached to a simple graph
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G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of
G. We provide sufficient conditions for Cohen–Macaulayness for closed and nonclosed graphs.
Binomial edge ideals arise naturally in the study of conditional independence ideals. Our 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. In this case the primary decomposition has a natural statistical interpretation.</abstract><cop>San Diego, CA</cop><pub>Elsevier Inc</pub><doi>10.1016/j.aam.2010.01.003</doi><tpages>17</tpages><oa>free_for_read</oa></addata></record> |
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| source | Elsevier ScienceDirect Journals; EZB-FREE-00999 freely available EZB journals |
| subjects | Algebra Binomial ideals Binomials Cohen–Macaulay rings Collection Commutative rings and algebras Conditional independence ideals Edge ideals Exact sciences and technology General mathematics General, history and biography Generators Graphs Mathematical analysis Mathematical models Mathematics Number theory Numerical analysis Numerical analysis. Scientific computation Radicals Robustness Sciences and techniques of general use |
| title | Binomial edge ideals and conditional independence statements |
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