Spectra of a 1-D wave equation on networks

By the graph theory, we first formulate the 1-D wave equation on networks in the form of a vector-valued differential equation in Cⁿ. Then we show that its spectra are the same as the zeros of an exponential polynomial, which implies that its spectra locate in a strip parallel to the imaginary axis in left half-plane. In the end, we calculate the spectral distribution of a star-shape network using the fortran package ZEAL to demonstrate our theoretical results.