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.


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.


  • Parry, W., & Sullivan, D. (1975). "A topological invariant of flows on 1-dimensional spaces". Topology. 14 (4): 297–299. doi:10.1016/0040-9383(75)90012-9.CS1 maint: multiple names: authors list (link)