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We study directed weighted graphs which are invariant under a nilpotent andcocompact group action. In particular, we consider the conic section K of theset of positive harmonic functions. We characterise the set of extreme pointsof the convex and compact set K as the set of multiplicative elements in K.Moreover, we study positive generalised eigenfunctions for a given parameter$\lambda$. We find that the topological space $M_\lambda$ of multiplicative$\lambda$-harmonic functions is homeomorphic to a sphere for $\lambda$ below acertain threshold.

Positive Harmonic Functions on Graphs with Nilpotent Group Actions