Witten-Morse functions and Morse inequalities on digraphs

TL;DR

This paper establishes a connection between discrete Morse functions on digraphs and Witten-Morse functions, demonstrating that their associated complexes approach Morse complexes and relating the homology to path homology.

Contribution

It introduces a new chain complex for digraphs that links critical paths to path homology and derives Morse inequalities for digraphs.

Findings

Witten complexes of transitive digraphs approach Morse complexes

Homology of the new chain complex is isomorphic to path homology

Morse inequalities are established for digraphs

Abstract

In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of paths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.