Mayer Path Homology

@misc{karaguler2026mayer,
  title = {{Mayer Path Homology}},
  author = {Dilan Karaguler and Guo-Wei Wei},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.16525},
  eprint = {2605.16525},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.16525},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Mayer Path Homology
AU  - Dilan Karaguler
AU  - Guo-Wei Wei
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2605.16525
UR  - https://arxiv.org/abs/2605.16525
AB  - 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 ∂-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.
ER  -