Abstract
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes, invariant frameworks and cyclic sequences. A set of discrete parameters for which there exist non-trivial solutions leads to the construction of a polynomial invariant and the notion of a geometric spectrum. Geometry then emerges, notably dimension, distance and curvature, from purely combinatorial properties of the graph.
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