Parry-Sullivan invariant

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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 matrices in graph theory, and of certain one-dimensional dynamical systems. It provides a partial classification of non-trivial irreducible incidence matrices.

It is named after the English mathematician Bill Parry and the American mathematician Dennis Sullivan, who introduced the invariant in a joint paper published in the journal Topology in 1975.

[edit] Definition

Let $A$ be an $n \times n$ incidence matrix. Then the Parry Sullivan number of $A$ is defined to be

PS(A) = det(I - A)

where $I$ denotes the $n \times n$ identity matrix.

[edit] 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.