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Nilpotent adjacency matrices, random graphs, and quantum random variables

Identifieur interne : 004584 ( Main/Merge ); précédent : 004583; suivant : 004585

Nilpotent adjacency matrices, random graphs, and quantum random variables

Auteurs : René Schott [France] ; Stacey Staples [États-Unis]

Source :

Journal of Physics A: Mathematical and Theoretical [ 1751-8113 ] ; 2008.

RBID : Hal:hal-00136290

Descripteurs français

mix :
- cycles, fermions, paths, quantum computing, random graphs.

Abstract

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.

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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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Nilpotent adjacency matrices, random graphs, and quantum random variables

Nilpotent adjacency matrices, random graphs, and quantum random variables

Source :

Descripteurs français

Abstract

Links toward previous steps (curation, corpus...)

Links to Exploration step

Le document en format XML

Pour manipuler ce document sous Unix (Dilib)

Pour mettre un lien sur cette page dans le réseau Wicri