Can you hear the size of the vertices? An inverse spectral problem of Laplacians on weighted graphs
McIlwain, Mary Hope
Doctor of Philosophy
Let G be a simple graph with n vertices. We define a Laplacian $\Delta$ on G which depends on an assignment of a weight to each vertex of G. One of the eigenvalues of $\Delta$ will always be 0. We fix the remaining (n $-$ 1) eigenvalues and ask for which graphs we can find a set of weights which generate a Laplacian with the spectrum $\Lambda.$ We demonstrate that it is always possible to solve the inverse spectral problem for a three vertex graph. In the case of $K\sb4$, the complete graph on four vertices, we prove several results. First, we give two proofs that it is always possible to solve the inverse problem. In addition, we give a description of all possible solution spaces for the inverse spectral problem. We also demonstrate that if we choose the eigenvalues to be positive it is not always possible to solve the inverse spectral problem with a set of positive weights.