  Spin ices can now be fabricated in a variety of geometries which control
their collective behavior and exotic properties. Therefore, their proper
framework is graph theory. We relate spin ice notions such as ice rule, ice
manifold, Coulomb phases, charges and monopoles, to graph-theoretical notions,
such as balance, in/out-degrees, and Eulerianicity. We then propose a
field-theoretical treatment in which topological charges and monopoles are the
degrees of freedom while the binary spins are subsumed into entropic
interaction among charges. We show that for a spin ice on a graph in a Gaussian
approximation, the kernel of the entropic interaction is the inverse of the
graph Laplacian, and we compute screening functions from the graph spectra as
Green operators for the screened Poisson problem on a graph. We then apply the
treatment on star graphs, tournaments, cycles, and regular spin ice in
different dimensions.
