A Description of the Parry-sullivan Number of a Graph Using Circuits

In this short note, we give a description of the Parry-Sullivan number of a graph in terms of the cycles in the graph. This tool is occasionally useful in reasoning about the Parry-Sullivan numbers of graphs. Given a graph E with n vertices, one may define the incidence matrix AE as the n × n matrix wherein each entry (AE)ij is defined to be the number of edges in E from the ith vertex to the jth vertex. Parry and Sullivan showed the quantity det(I − AE), now known as the Parry-Sullivan number of the graph and denoted PS(E), is an invariant of the flow equivalence class of the subshift of finite type induced by E [1]. (Equivalently, some sources choose to view PS(E) as det(I − AtE).) In working with this invariant, it is sometimes convenient to view the Parry-Sullivan number of a graph in terms of the structure of cycles of the graph, rather than as a determinant calculation. We establish such a characterization here. Some basic definitions are given here and used throughout this note. Definition 1. A graph E (also known as a directed graph) is a 4-tuple (E, E, r, s), where E is a set of vertices, E is a set of edges, and r, s : E → E associate each edge with its range and source, respectively. • A directed cycle of a graph E is a sequence of edges e1e2 · · · em such that s(e1) = r(em), and s(ei) 6= s(ej) whenever i 6= j. In particular, the choice of starting vertex, s(e1), distinguishes two cycles that follow the same edges. • A directed circuit in a graph E is a finite set of edges C ⊆ E with the property that the edges in C can be arranged into a directed cycle. • Let C be a set of directed circuits. C is vertex-disjoint if for any C1, C2 ∈ C, the sets {s(e)|e ∈ C1} and {s(e)|e ∈ C2} are disjoint. In this note, by the words cycle and circuit, we always mean directed cycles and directed circuits. We now view circuits as inducing a permutation on the vertices of a graph, in the obvious way: Definition 2. Let C be a set of vertex-disjoint circuits of a graph E. The permutation ρC : E 0 → E induced by C is defined as follows. If there exists e ∈ ⋃ C such that s(e) = v, then ρC(v) = r(e). Otherwise, ρC(v) = v. Then ρC is well-defined because C is vertex-disjoint, and it is easily seen to be a permutation of E. The curcuits having more than one edge in C correspond with cycles in the representation of ρC as a product of disjoint cycles. Several distinct such vertex-disjoint sets of curcuits may induce the same permutation of E. The induced permutation ρC can be viewed as defining an equivalence relation on vertex-disjoint sets of circuits, as follows: 1 Definition 3. The equivalence relation ∼ρ is defined such that C1 ∼ρ C2 if and only if ρC1 = ρC2 . A quick lemma about permutations will be useful later. Lemma 4. Let ρ ∈ Sn. If ρ is the product of m disjoint cycles σ1σ2 · · ·σm, and ρ has k fixed points, then ρ is a product of n− (m+ k) transpositions. Proof. A cycle of length j can be obtained by a product of j−1 transpositions. By repeating this for each disjoint cycle in ρ, one can express ρ as the product of ∑m i=1(|σi| − 1) transpositions. But each 1 ≤ a ≤ n is either a member of some σi or is a fixed point of ρ, so ∑m i=1 |σi| = n− k, and rearranging the sum gives ρ as a product of m ∑