Given two dual graphs G and G* embedded to a manifold M(G), for every harmonic function u, defined on the vertices of G, one can define a harmonic conjugate v of u on the vertices of G* using the analogy of Cauchy-Riemann equations. In the case of a simply connected region of the complex plane C the harmonic conjugate is well-defined up to an additive constant.
Exercise (*). Prove that a conjugation is a duality relationship on the space of harmonic functions on dual planar graphs.
The Dirichlet-to-Neumann operator of a planar graph essentially gives the correspondence between boundary values of a harmonic function and its conjugate.