  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.
