Abstract
A Riemann-Roch theorem on graph was initiated by M.Baker and S.Norine. In their article, a Riemann-Roch theorem on a finite graph with uniform unit vertex-weight and uniform unit edge-weight was established and a feasibility of Riemann-Roch theorem on infinite graph was suggested. 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 \(\mathbb Q\)-valued positive weights on the edges. We build a potential theoretic scheme for a proof of a Riemann-Roch theorem on the edge-weighted infinite graphs.
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Atsuji, A., Kaneko, H. (2021). A Riemann-Roch Theorem on Infinite Graphs. In: Zúñiga-Galindo, W.A., Toni, B. (eds) Advances in Non-Archimedean Analysis and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-030-81976-7_9
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DOI: https://doi.org/10.1007/978-3-030-81976-7_9
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