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

The Concept of Spin Ice Graphs and a Field Theory for their Topological Monopoles and Charges