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.