Energy integral sum

The energy integral represents the area of the image of the unique conformal map (discrete analytic map) under harmonic function with given boundary conditions: and for the dual graph/network harmonic conjugate: For the analytic contimuation of Laplacian The harmonic function minimizes the integral with given boundary conditions and the response Dirichlet-to-Neumann map sends the boundary potential to boundary current for the harmonic function in interior...

For and, therefore Laplace, is Hermitian with non-negative eigenvalues.

We study the zeros and poles (monodromy eigenvalues) of the Schur complement of . They're simple positive and interlace for the circular planar graph because of total positivity of the monodromy matrix in the alternating basis...

Following the cycles in the Spiderweb network: , w is the N'th root of unity, where the roots 's are positive numbers (reciprocal eigenvalues of monodromy matrix) interlacing for homogenuous Dirichlet and Neumann boundary conditions... The continuous differential version of

is generalization of , fractal Laplace operator in polar coordinates.