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A salami is a connected, locally finite, weighted graph with non-negativeOllivier Ricci curvature and at least two ends of infinite volume. We show thatevery salami has exactly two ends and no vertices with positive curvature. Wemoreover show that every salami is recurrent and admits harmonic functions withconstant gradient. The proofs are based on extremal Lipschitz extensions, avariational principle and the study of harmonic functions. Assuming a lowerbound on the edge weight, we prove that salamis are quasi-isometric to theline, that the space of all harmonic functions has finite dimension, and thatthe space of subexponentially growing harmonic functions is two-dimensional.Moreover, we give a Cheng-Yau gradient estimate for harmonic functions onballs.

Every Salami has two ends