Minimal Path and Acyclic Models in the Path Complex

TL;DR

This paper investigates the structure of path complexes in directed graphs, focusing on minimal paths and their acyclic properties, with applications to understanding the homology of such models.

Contribution

It introduces a detailed study of minimal paths in path complexes and demonstrates their acyclic homologies, expanding the theoretical framework of digraphs.

Findings

Minimal paths are characterized under strongly regular conditions.

Supporting sub-digraphs of minimal paths have acyclic homologies.

Applications of acyclic models are discussed.

Abstract

In this paper, firstly, we will study the structure of the path complex $(\Omega_*(G;\Z),\partial)$ of a digraph $G$ via the $\Z$-generators of $\Omega_*(G,\Z)$ under strongly regular condition, which is called the minimal path in \cite{HY}. In particular, we will study various examples of the minimal $3$-paths. Secondly, we will show that the supporting sub-digraph of minimal path has acyclic path homologies. Thirdly, we will consider the applications of such an acyclic model.