Nilpotent adjacency matrices, random graphs and quantum random variables
Identifieur interne : 001798 ( Istex/Curation ); précédent : 001797; suivant : 001799Nilpotent 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.
English descriptors
- 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:
DOI: 10.1088/1751-8113/41/15/155205
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<term>Annihilation operators</term>
<term>Canonical</term>
<term>Clifford algebras</term>
<term>Commutation relations</term>
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<term>Degree operator</term>
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<term>Fermion adjacency operator</term>
<term>Fermion annihilation operators</term>
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<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>
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<term>Phys</term>
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<term>Random graphs</term>
<term>Random variables</term>
<term>Schott</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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