A family of partitions of the set of walks on a directed graph

Abstract

We present a family of partitions of $W_\mathcal{G}$, the set of walks on a directed graph $\mathcal{G}$. Each partition in this family is identified by an integer sequence $K$, which specifies a collection of cycles on $\mathcal{G}$ with a certain well-defined structure. We term such cycles resummable, and a walk that does not traverse any such cycles $K$-irreducible. For a given value of $K$, the corresponding partition of $W_\mathcal{G}$ consists of a collection of cells that each contain a single $K$-irreducible walk $i$ plus all walks that can be formed from $i$ by attaching one or more resummable cycles to its vertices. We characterise the entire family of partitions of $W_\mathcal{G}$ by giving explicit expressions for the structure of the $K$-irreducible walks and the resummable cycles for arbitrary values of $K$. We demonstrate how these results can be exploited to recast the sum over all walks on a directed graph as a sum over dressed $K$-irreducible walks, and discuss the applications of this reformulation to matrix computations.