Cospectral graphs obtained by edge deletion

We show how to construct graphs which are simultaneously cospectral with respect to several symmetric matrices that encode adjacency. Our main operation consists of locating cliques in highly regular graphs (for example, strongly regular graphs) and removing edges in distinct ways preserving the spectrum of the adjacency, Laplacian, unsigned Laplacian, and normalized Laplacian matrix. We also provide a more general construction of Laplacian cospectral graphs. This has applications in Laplacian state transfer in the study of quantum information transfer.