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arXiv · 2201.07710

A Riemann-Roch theorem on a weighted infinite graph

Atsushi Atsuji·Hiroshi Kaneko
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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.

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Atsushi Atsuji, Hiroshi Kaneko. 2022-01-19. A Riemann-Roch theorem on a weighted infinite graph. https://arxiv.org/abs/2201.07710

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