User:Daviddaved/One more graph example

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.

Last modified on 4 October 2012, at 01:00