Let $R$ be any subring of the reals. We present a generalization of linear systems on graphs where divisors are $R$-valued functions on the set of vertices and graph edges are permitted to have nonnegative weights in $R$. Using this generalization, we provide an independent proof of a Riemann-Roch formula, which implies the Riemann-Roch formula of Baker and Norine.
"Linear systems on edge-weighted graphs." Rocky Mountain J. Math. 46 (5) 1559 - 1574, 2016. https://doi.org/10.1216/RMJ-2016-46-5-1559