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.
Let A be an n × n incidence matrix. Then the Parry–Sullivan number of A is defined to be
where I denotes the n × n identity matrix.
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.