Mayer Path Homology

TL;DR

Mayer path homology introduces a new $N$-nilpotent differential for directed path complexes, providing a more sensitive invariant for directed graphs and revealing novel algebraic structures.

Contribution

This work develops Mayer path homology with an $N$-differential, offering a systematic structural theory and enhanced graph invariants beyond standard path homology.

Findings

Defines Mayer path homology as a canonical directed graph invariant.

Distinguishes directed network motifs not separable by standard path homology.

Classifies generators of $ abla_2^N$ and $ abla_3^N$, characterizing admissible types.

Abstract

We introduce Mayer path homology, a new homology theory for directed path complexes obtained by equipping path complexes with an $N$-nilpotent differential. The main novelty of this work is the introduction of an $N$-differential on path complexes, giving rise to $N$-chain complexes of $\partial$-invariant paths and Mayer path homology groups $H_n^{N,q}(P)$. We prove that this construction defines a canonical invariant of directed graphs and is more sensitive than standard path homology, distinguishing directed network motifs that ordinary path homology cannot separate. We further establish a complete classification of generators of $\Omega_2^N$ and $\Omega_3^N$, determining all admissible combinatorial types. Finally, we characterize elements of the first Mayer path cycles group $Z_1^{N,q}$ in terms of weighted directed cycles arising from spanning-tree constructions. These results provide the first systematic structural theory for Mayer path complexes and reveal new higher-order algebraic structures in directed graphs.