Numerical analysis of scattering on combinatorial graphs

Abstract

We investigate numerically the scattering of waves on discrete graphs. An efficient algorithm is developed to compute the reflection and transmission (spectral) coefficients. We then explore various configurations of input and output leads, demonstrating how bound states arising from specific vertex-lead connections result in total reflection. The impedance of the leads is shown to influence the spectral coefficients in a predictable manner. Furthermore, for a given input lead we show that the total transmission of a wave packet can be maximized by appropriately selecting the exit lead. Finally, we analyze the spectral signatures of defects within the graph and find that they vary depending on both the defect's location and its spectral characteristics thus enabling its identification.