Signed graphs whose all Laplacian eigenvalues are main

For a graph G we consider the problem of the existence of a switching equivalent signed graph with Laplacian eigenvalues that are all main and the problem of determination of all switching equivalent signed graphs with this spectral property. Using a computer search we confirm that apart from K-2 every connected graph with at most 7 vertices switches to at least one signed graph with the required property. This fails to hold for exactly 22 connected graphs with 8 vertices. If G is a cograph without repeated eigenvalues, then we give an iterative solution for the latter problem and the complete solution in the particular case when G is a threshold graph. The first problem is resolved positively for a particular class of chain graphs. The obtained results are applicable in control theory for generating controllable signed graphs based on Laplacian dynamics.