The following construction provides an example of an infinite graph, Dirichlet-to-Neumann operator of which satisfies the operator equation in the title of this chapter.

The operator equation reflects the self-duality and self-symmetry of the infinite graph.

**Exercise (**).** Prove that the Dirichlet-to-Neumann operator of the graph with the natural boundary satisfies the functional equation. (Hint) Use the fact that the operator/matrix is the fixed point of the Schur complement

where

is the circular matrix of first differences.