Homologies of path complexes and digraphs

TL;DR

This paper introduces path complexes as a generalization of simplicial complexes, motivated by directed graphs, and develops a new framework for path homology and cohomology of digraphs.

Contribution

It proposes a novel concept of path complexes and establishes a new homology and cohomology theory for directed graphs, extending existing topological methods.

Findings

Defined path complexes as a generalization of simplicial complexes

Developed a new homology and cohomology theory for digraphs

Provided foundational results connecting path complexes to directed graph topology

Abstract

In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs(digraphs). We obtain a new notion of the path homology and cohomology of a digraph.