Parry-Sullivan invariant

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.

Definition

Let "A" be an "n" × "n" incidence matrix. Then the Parry-Sullivan number of "A" is defined to be

:$mathrm\{PS\}\; (A)\; =\; det\; (I\; -\; A)$,

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

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