Constructing Cospectral Vertices Through Orbits of Subgraphs

A constructive method is given for obtaining cospectral vertices in undirected graphs, along with an operation that preserves this construction. We prove that the construction yields cospectral vertices, as well as strongly cospectral vertices under additional conditions. Furthermore, we generalize cospectral vertices to the case of the graph Laplacian and provide an analogous construction.