Nilpotent adjacency matrices, random graphs and quantum random variables
Identifieur interne : 004349 ( Main/Exploration ); précédent : 004348; suivant : 004350Nilpotent adjacency matrices, random graphs and quantum random variables
Auteurs : Ren Schott [France] ; George Stacey Staples [États-Unis]Source :
- Journal of Physics A: Mathematical and Theoretical [ 1751-8113 ] ; 2008.
Descripteurs français
- Pascal (Inist)
- mix :
English descriptors
- KwdEn :
- Teeft :
- Adjacency, Adjacency matrix, Algebra, Algebraic probability space, Annihilation operators, Canonical, Clifford algebras, Commutation relations, Computational complexity, Degree operator, Fermion, Fermion adjacency operator, Fermion annihilation operators, Fock, Fock spaces, Graph, Graph theory, Hilbert, Hilbert space, Initial vertex, Inner product, Kronecker delta function, Markov, Markov chain, Markov chains, Matrix, Nilpotent adjacency matrices, Nilpotent adjacency operator, Nilpotent adjacency operators, Nite, Nite graph, Nite graphs, Nonzero terms, Obata, Other words, Phys, Positive integer, Probability space, Quantum, Quantum decomposition, Quantum probab, Quantum probability, Random graphs, Random variables, Schott, Staple, Star graphs, Terminal vertex, Theor, Transition matrix.
Abstract
While a number of researchers have previously investigated the relationship between graph theory and quantum probability, the current work explores a new perspective. The approach of this paper is to begin with an arbitrary graph having no previously established relationship to quantum probability and to use that graph to construct a quantum probability space in which moments of quantum random variables reveal information about the graph's structure. Given an arbitrary finite graph and arbitrary odd integer m 3, fermion annihilation operators are used to construct a family of quantum random variables whose mth moments correspond to the graph's m-cycles. The approach is then generalized to recover a graph's m-cycles for any integer m 3 by defining nilpotent adjacency operators in terms of null-square generators of an infinite-dimensional Abelian algebra. It is shown that ordering the vertices of a simple graph induces a canonical decomposition on any nilpotent adjacency operator . The work concludes with applications to Markov chains and random graphs.
Url:
- https://api.istex.fr/ark:/67375/0T8-PJZXK80J-S/fulltext.pdf
- https://hal.archives-ouvertes.fr/hal-00136290
DOI: 10.1088/1751-8113/41/15/155205
Affiliations:
Links toward previous steps (curation, corpus...)
- to stream Istex, to step Corpus: 001817
- to stream Istex, to step Curation: 001798
- to stream Istex, to step Checkpoint: 000D77
- to stream Main, to step Merge: 004460
- to stream Hal, to step Corpus: 006C47
- to stream Hal, to step Curation: 006C47
- to stream Hal, to step Checkpoint: 003468
- to stream Main, to step Merge: 004584
- to stream PascalFrancis, to step Corpus: 000315
- to stream PascalFrancis, to step Curation: 000712
- to stream PascalFrancis, to step Checkpoint: 000272
- to stream Main, to step Merge: 004567
- to stream Main, to step Curation: 004349
Le document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Nilpotent adjacency matrices, random graphs and quantum random variables</title><author><name sortKey="Schott, Ren" sort="Schott, Ren" uniqKey="Schott R" first="Ren" last="Schott">Ren Schott</name></author><author><name sortKey="Staples, George Stacey" sort="Staples, George Stacey" uniqKey="Staples G" first="George Stacey" last="Staples">George Stacey Staples</name></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:67DF8C7FB0CA6E7547CE9339BF63F4AE20BCF722</idno><date when="2008" year="2008">2008</date><idno type="doi">10.1088/1751-8113/41/15/155205</idno><idno type="url">https://api.istex.fr/ark:/67375/0T8-PJZXK80J-S/fulltext.pdf</idno><idno type="wicri:Area/Istex/Corpus">001817</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001817</idno><idno type="wicri:Area/Istex/Curation">001798</idno><idno type="wicri:Area/Istex/Checkpoint">000D77</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Checkpoint">000D77</idno><idno type="wicri:doubleKey">1751-8113:2008:Schott R:nilpotent:adjacency:matrices</idno><idno type="wicri:Area/Main/Merge">004460</idno><idno type="wicri:source">HAL</idno><idno type="RBID">Hal:hal-00136290</idno><idno type="url">https://hal.archives-ouvertes.fr/hal-00136290</idno><idno type="wicri:Area/Hal/Corpus">006C47</idno><idno type="wicri:Area/Hal/Curation">006C47</idno><idno type="wicri:Area/Hal/Checkpoint">003468</idno><idno type="wicri:explorRef" wicri:stream="Hal" wicri:step="Checkpoint">003468</idno><idno type="wicri:doubleKey">1751-8113:2008:Schott R:nilpotent:adjacency:matrices</idno><idno type="wicri:Area/Main/Merge">004584</idno><idno type="wicri:source">INIST</idno><idno type="RBID">Pascal:08-0216287</idno><idno type="wicri:Area/PascalFrancis/Corpus">000315</idno><idno type="wicri:Area/PascalFrancis/Curation">000712</idno><idno type="wicri:Area/PascalFrancis/Checkpoint">000272</idno><idno type="wicri:explorRef" wicri:stream="PascalFrancis" wicri:step="Checkpoint">000272</idno><idno type="wicri:doubleKey">1751-8113:2008:Schott R:nilpotent:adjacency:matrices</idno><idno type="wicri:Area/Main/Merge">004567</idno><idno type="wicri:Area/Main/Curation">004349</idno><idno type="wicri:Area/Main/Exploration">004349</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Nilpotent adjacency matrices, random graphs and quantum random variables</title><author><name sortKey="Schott, Ren" sort="Schott, Ren" uniqKey="Schott R" first="Ren" last="Schott">Ren Schott</name><affiliation wicri:level="3"><country xml:lang="fr">France</country><wicri:regionArea>IECN and LORIA Universit Henri Poincar-Nancy I, BP 239, 54506 Vandoeuvre-ls-Nancy</wicri:regionArea><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandoeuvre-ls-Nancy</settlement></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">France</country></affiliation></author><author><name sortKey="Staples, George Stacey" sort="Staples, George Stacey" uniqKey="Staples G" first="George Stacey" last="Staples">George Stacey Staples</name><affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653</wicri:regionArea><placeName><region type="state">Illinois</region></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">États-Unis</country></affiliation></author></analytic><monogr></monogr><series><title level="j">Journal of Physics A: Mathematical and Theoretical</title><title level="j" type="abbrev">J. Phys. A: Math. Theor.</title><idno type="ISSN">1751-8113</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2008">2008</date><biblScope unit="volume">41</biblScope><biblScope unit="issue">15</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="16">16</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint><idno type="ISSN">1751-8113</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">1751-8113</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Annihilation operators</term><term>Fermions</term><term>Finite graph</term><term>Infinite dimension</term><term>Markov chain</term><term>Probability</term><term>Quantum graph</term><term>Quantum theory</term><term>Random graph</term><term>Random matrix</term><term>Random variable</term><term>Vertex</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Chaîne Markov</term><term>Dimension infinie</term><term>Fermion</term><term>Graphe aléatoire</term><term>Graphe fini</term><term>Graphe quantique</term><term>Matrice aléatoire</term><term>Opérateur annihilation</term><term>Probabilité</term><term>Théorie quantique</term><term>Variable aléatoire</term><term>Vertex</term></keywords><keywords scheme="Teeft" xml:lang="en"><term>Adjacency</term><term>Adjacency matrix</term><term>Algebra</term><term>Algebraic probability space</term><term>Annihilation operators</term><term>Canonical</term><term>Clifford algebras</term><term>Commutation relations</term><term>Computational complexity</term><term>Degree operator</term><term>Fermion</term><term>Fermion adjacency operator</term><term>Fermion annihilation operators</term><term>Fock</term><term>Fock spaces</term><term>Graph</term><term>Graph theory</term><term>Hilbert</term><term>Hilbert space</term><term>Initial vertex</term><term>Inner product</term><term>Kronecker delta function</term><term>Markov</term><term>Markov chain</term><term>Markov chains</term><term>Matrix</term><term>Nilpotent adjacency matrices</term><term>Nilpotent adjacency operator</term><term>Nilpotent adjacency operators</term><term>Nite</term><term>Nite graph</term><term>Nite graphs</term><term>Nonzero terms</term><term>Obata</term><term>Other words</term><term>Phys</term><term>Positive integer</term><term>Probability space</term><term>Quantum</term><term>Quantum decomposition</term><term>Quantum probab</term><term>Quantum probability</term><term>Random graphs</term><term>Random variables</term><term>Schott</term><term>Staple</term><term>Star graphs</term><term>Terminal vertex</term><term>Theor</term><term>Transition matrix</term></keywords><keywords scheme="mix" xml:lang="fr"><term>cycles</term><term>fermions</term><term>paths</term><term>quantum computing</term><term>random graphs</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">While a number of researchers have previously investigated the relationship between graph theory and quantum probability, the current work explores a new perspective. The approach of this paper is to begin with an arbitrary graph having no previously established relationship to quantum probability and to use that graph to construct a quantum probability space in which moments of quantum random variables reveal information about the graph's structure. Given an arbitrary finite graph and arbitrary odd integer m 3, fermion annihilation operators are used to construct a family of quantum random variables whose mth moments correspond to the graph's m-cycles. The approach is then generalized to recover a graph's m-cycles for any integer m 3 by defining nilpotent adjacency operators in terms of null-square generators of an infinite-dimensional Abelian algebra. It is shown that ordering the vertices of a simple graph induces a canonical decomposition on any nilpotent adjacency operator . The work concludes with applications to Markov chains and random graphs.</div></front></TEI><affiliations><list><country><li>France</li><li>États-Unis</li></country><region><li>Grand Est</li><li>Illinois</li><li>Lorraine (région)</li></region><settlement><li>Vandoeuvre-ls-Nancy</li></settlement></list><tree><country name="France"><region name="Grand Est"><name sortKey="Schott, Ren" sort="Schott, Ren" uniqKey="Schott R" first="Ren" last="Schott">Ren Schott</name></region><name sortKey="Schott, Ren" sort="Schott, Ren" uniqKey="Schott R" first="Ren" last="Schott">Ren Schott</name></country><country name="États-Unis"><region name="Illinois"><name sortKey="Staples, George Stacey" sort="Staples, George Stacey" uniqKey="Staples G" first="George Stacey" last="Staples">George Stacey Staples</name></region><name sortKey="Staples, George Stacey" sort="Staples, George Stacey" uniqKey="Staples G" first="George Stacey" last="Staples">George Stacey Staples</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Lorraine/explor/InforLorV4/Data/Main/Exploration
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 004349 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 004349 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien
|wiki= Wicri/Lorraine
|area= InforLorV4
|flux= Main
|étape= Exploration
|type= RBID
|clé= ISTEX:67DF8C7FB0CA6E7547CE9339BF63F4AE20BCF722
|texte= Nilpotent adjacency matrices, random graphs and quantum random variables
}}
| This area was generated with Dilib version V0.6.33. |  |