# 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[edit]

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.

## Properties[edit]

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[edit]

- 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)