# Numerical analysis of scattering on combinatorial graphs

## TL;DR

This paper presents a numerical study of wave scattering on discrete graphs, introducing an efficient algorithm to analyze spectral coefficients, and explores how graph configurations affect wave transmission, reflection, and defect detection.

## Contribution

It develops a novel efficient algorithm for computing spectral scattering coefficients on graphs and investigates how graph structure influences wave behavior and defect identification.

## Key findings

- Bound states cause total reflection in certain configurations
- Lead impedance affects spectral coefficients predictably
- Wave packet transmission can be maximized by lead selection

## 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.