A Riemann-Roch theorem on a weighted infinite graph
Combinatorics
2022-01-20 v1 Probability
Abstract
A Riemann-Roch theorem on graph was initiated by M. Baker and S. Norine. In their article [2], a Riemann-Roch theorem on a finite graph with uniform vertex-weight and uniform edge-weight was established and it was suggested a Riemann-Roch theorem on an infinite graph was feasible. In this article, we take an edge-weighted infinite graph and focus on the importance of the spectral gaps of the Laplace operators defined on its finite subgraphs naturally given by Q-valued positive weights on the edges. We build a potential theoretic scheme for proof of a Riemann-Roch theorem on the edge-weighted infinite graph.
Cite
@article{arxiv.2201.07710,
title = {A Riemann-Roch theorem on a weighted infinite graph},
author = {Atsushi Atsuji and Hiroshi Kaneko},
journal= {arXiv preprint arXiv:2201.07710},
year = {2022}
}