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Recovering Laplacian Lattices from L-Functions of Graphs

  • Daniel Labib
  • Antonio Lei
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Abstract

We introduce L-functions associated with characters of the Jacobian of a finite graph, as a graph-theoretic analogue of the L-functions arising from unramified coverings of algebraic curves. These L-functions are defined using the Riemann--Roch structure on the graph and extend Lorenzini's two-variable zeta function. We show that if two connected graphs without bridges have isomorphic Jacobians and their L-functions agree under the induced correspondence of characters, then their Laplacian lattices coincide. We also show that these L-functions satisfy a functional equation. Finally, we give examples showing that neither the Jacobian nor the Lorenzini zeta function alone determines the Laplacian lattice.

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Cite

BibTeXRIS
BibTeX
@misc{labib2026recovering,
  title = {{Recovering Laplacian Lattices from $L$-Functions of Graphs}},
  author = {Daniel Labib and Antonio Lei},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.29981},
  eprint = {2608.29981},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29981},
}
RIS
TY  - UNPB
TI  - Recovering Laplacian Lattices from L-Functions of Graphs
AU  - Daniel Labib
AU  - Antonio Lei
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2608.29981
UR  - https://arxiv.org/abs/2608.29981
AB  - We introduce L-functions associated with characters of the Jacobian of a finite graph, as a graph-theoretic analogue of the L-functions arising from unramified coverings of algebraic curves. These L-functions are defined using the Riemann--Roch structure on the graph and extend Lorenzini's two-variable zeta function. We show that if two connected graphs without bridges have isomorphic Jacobians and their L-functions agree under the induced correspondence of characters, then their Laplacian lattices coincide. We also show that these L-functions satisfy a functional equation. Finally, we give examples showing that neither the Jacobian nor the Lorenzini zeta function alone determines the Laplacian lattice.
ER  -