In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of
incidence matrices in
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, and of certain one-dimensional
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
. It provides a partial classification of non-trivial irreducible incidence matrices.
It is named after the English mathematician
Bill Parry William or Bill Parry may refer to:
Sports
*William Parry Crake (1852–1921), or William Parry, Wanderers footballer
*Bill Parry (footballer, born 1873) (1873–1923), Welsh international footballer
*Bill Parry (footballer, born 1914) (1914–196 ...
and the American mathematician
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Ce ...
, who introduced the invariant in a joint paper published in the journal
''Topology'' in 1975.
Definition
Let ''A'' be an ''n'' × ''n''
incidence matrix
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element ...
. Then the Parry–Sullivan number of ''A'' is defined to be
:
where ''I'' denotes the ''n'' × ''n'' identity matrix.
Properties
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.
References
{{DEFAULTSORT:Parry-Sullivan invariant
Dynamical systems
Matrices
Algebraic graph theory
Graph invariants