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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.16525 |
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Table of Contents:
- 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 $Ω_2^N$ and $Ω_3^N$, 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.