# The maximal subgroups and the complexity of the flow semigroup of finite (di)graphs

## TL;DR

This paper refines the understanding of the structure of flow semigroups in finite (di)graphs, proves Rhodes's conjecture for certain cases, and provides an efficient algorithm to compute defect groups.

## Contribution

It proves Rhodes's conjecture on the structure of maximal groups in flow semigroups for finite, antisymmetric, strongly connected digraphs and describes the actions of these subgroups.

## Key findings

- Confirmed Rhodes's conjecture for 2-vertex connected strongly connected graphs with n vertices.
- Developed a linear algorithm to determine defect k groups for any finite (di)graph.
- Fully described the structure and actions of maximal subgroups acting on all but k points.

## Abstract

The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. After collecting together previous partial results, we refine and prove Rhodes's conjecture on the structure of the maximal groups in the flow semigroup for finite, antisymmetric, strongly connected digraphs. Building on this result, we investigate and fully describe the structure and actions of the maximal subgroups of the flow semigroup acting on all but $k$ points for all finite digraphs and graphs for all $k\geq 1$. A linear algorithm (in the number of edges) is presented to determine these so-called `defect $k$ groups' for any finite (di)graph. Finally, we prove that the complexity of the flow semigroup of a 2-vertex connected (and strongly connected di)graph with $n$ vertices is $n-2$, completely confirming Rhodes's conjecture for such (di)graphs.