  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.
