A Riemann-Roch theorem on a weighted infinite graph

TL;DR

This paper extends the Riemann-Roch theorem to infinite weighted graphs by analyzing spectral gaps of Laplace operators on finite subgraphs and developing a potential theoretic proof scheme.

Contribution

It introduces a Riemann-Roch theorem for infinite edge-weighted graphs using spectral analysis and potential theory, expanding prior finite graph results.

Findings

Established a Riemann-Roch theorem for infinite weighted graphs.

Linked spectral gaps of Laplace operators to graph properties.

Developed a potential theoretic proof approach.

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.