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| In [[mathematics]], the '''Parry–Sullivan invariant''' (or '''Parry–Sullivan number''') is a numerical quantity of interest in the study of [[incidence matrix|incidence matrices]] in [[graph theory]], and of certain one-dimensional [[dynamical systems]]. It provides a partial classification of non-trivial irreducible incidence matrices.
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| It is named after the English mathematician [[Bill Parry (mathematician)|Bill Parry]] and the American mathematician [[Dennis Sullivan]], who introduced the invariant in a joint paper published in the journal [[Topology (journal)|''Topology'']] in 1975.
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| ==Definition==
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| Let ''A'' be an ''n'' × ''n'' incidence matrix. Then the '''Parry–Sullivan number''' of ''A'' is defined to be
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| :<math>\mathrm{PS} (A) = \det (I - A), \, </math>
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| where ''I'' denotes the ''n'' × ''n'' identity matrix.
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| ==Properties==
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| It can be shown that, for nontrivial irreducible incidence matrices, flow equivalence is completely determined by the Parry–Sullivan number and the [[Bowen–Franks group]].
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| ==References==
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| * {{cite journal | author=Parry, W., & Sullivan, D. | title=A topological invariant of flows on 1-dimensional spaces | journal=Topology | volume=14 | year=1975 | pages=297–299 | doi=10.1016/0040-9383(75)90012-9 | issue=4}}
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| {{DEFAULTSORT:Parry-Sullivan invariant}}
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| [[Category:Dynamical systems]]
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| [[Category:Matrices]]
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| [[Category:Algebraic graph theory]]
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| [[Category:Graph invariants]]
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Latest revision as of 04:36, 13 October 2014
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