Exterior-Interior Duality for Discrete Graphs

TL;DR

This paper extends the exterior-interior duality, originally in continuous domains, to the discrete setting of finite graphs by developing two methods to associate scattering matrices and establishing the duality for both.

Contribution

It introduces two novel methods for associating scattering matrices to finite graphs and extends the exterior-interior duality to the discrete Laplacian spectrum.

Findings

Established duality for both methods

Connected scattering matrices to graph spectra

Provided new tools for spectral analysis of graphs

Abstract

The Exterior-Interior duality expresses a deep connection between the Laplace spectrum in bounded and connected domains in $\mathbb{R}^2$, and the scattering matrices in the exterior of the domains. Here, this link is extended to the study of the spectrum of the discrete Laplacian on finite graphs. For this purpose, two methods are devised for associating scattering matrices to the graphs. The Exterior -Interior duality is derived for both methods.