  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.
