# User:Daviddaved/One more graph example

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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.

$\Lambda(G) = \sqrt{L}.$

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

The self-dual and self-symmetric 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

$\Lambda(G) = \begin{pmatrix} 2I & B \\ B^T & \Lambda + 2I \end{pmatrix}/ (\Lambda + 2I),$

where

$B = \begin{pmatrix} -1 & 0 & 0 & \ldots & -1 \\ -1 & -1 & 0 & \ldots & 0 \\ 0 & \vdots & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & -1 & 0 \\ 0 & 0 & \ldots & -1 & -1 \\ \end{pmatrix}$

is the circular matrix of first differences.