Using the Laplacian Matrix as a Differential Operator in Quantum Graphs

A quantum graph is a weighted combinatorial graph equipped with a Hamiltonian operator acting on functions along each edge. As the name suggests, it can be used to model quantum phenomena such as wave propagation and free-electron theory. We investigate the behavior of quantum graphs when their Laplacian matrices are used as the operator. Specifically, we examine families of graphs and solve for their characteristic functions and vertex conditions based on the eigenvalues of the Laplacian. Preliminary results suggest a systematic method for complete graphs, star graphs, path graphs, and cycles. This would provide some of the first results on the relation between spectral graph theory and quantum graphs