InforLorV4, Main, Curation, bibRecord, 004349

Nilpotent adjacency matrices, random graphs and quantum random variables

Identifieur interne : 004349 ( Main/Curation ); précédent : 004348; suivant : 004350

Nilpotent 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.

RBID : ISTEX:67DF8C7FB0CA6E7547CE9339BF63F4AE20BCF722

Descripteurs français

Pascal (Inist)
- Chaîne Markov, Dimension infinie, Fermion, Graphe aléatoire, Graphe fini, Graphe quantique, Matrice aléatoire, Opérateur annihilation, Probabilité, Théorie quantique, Variable aléatoire, Vertex.
mix :
- cycles, fermions, paths, quantum computing, random graphs.

English descriptors

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:

DOI: 10.1088/1751-8113/41/15/155205

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Le document en format XML

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<front><div type="abstract" xml:lang="en">For fixed $n>0$, the space of finite graphs on $n$ vertices is canonically associated with an abelian, nilpotent-generated subalgebra of the $2n$-particle fermion algebra. using the generators of the subalgebra, an algebraic probability space of "nilpotent adjacency matrices" associated with finite graphs is defined. Each nilpotent adjacency matrix is a quantum random variable whose $m^th$ moment corresponds to the number of $m$-cycles in the graph $G$. Each matrix admits a canonical "quantum decomposition" into a sum of three algebraic random variables: $a = a^\Delta+ a^\Upsilon+a^Lambda$, where $a^\Delta$ is classical while $a^\Upsilon and $a^\Lambda$ are quantum. Moreover, within the algebraic context, the NP problem of cycle enumeration is reduced to matrix multiplication, requiring no more than $n^4$ multiplications within the algebra.</div>
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<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>
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<term>Graphe aléatoire</term>
<term>Graphe quantique</term>
<term>Variable aléatoire</term>
<term>Théorie quantique</term>
<term>Probabilité</term>
<term>Graphe fini</term>
<term>Fermion</term>
<term>Opérateur annihilation</term>
<term>Dimension infinie</term>
<term>Vertex</term>
<term>Chaîne Markov</term>
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<front><div type="abstract" xml:lang="en">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 decompositionΨ =Ψ<sup>+</sup>
 + Ψ<sup>-</sup>
 on any nilpotent adjacency operator Ψ. The work concludes with applications to Markov chains and random graphs.</div>
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<profileDesc><textClass><keywords scheme="Teeft" xml:lang="en"><term>Adjacency</term>
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<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>
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<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>
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Nilpotent adjacency matrices, random graphs and quantum random variables

Nilpotent adjacency matrices, random graphs and quantum random variables

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