Second laziest graph for alternative graph matrices

Determine whether the star S_n is the second laziest connected graph on n vertices when laziness is defined relative to the adjacency matrix, the unsigned Laplacian matrix, and the normalized Laplacian matrix.

Background

For Laplacian quantum walks, the paper proves that the star S_n is the second laziest connected graph on n vertices. It proposes testing whether the same ordering persists when the walk and associated average mixing matrix are defined using other standard graph matrices. Computational data for graphs with up to nine vertices is cited as motivation, but the question is left unresolved for the adjacency, unsigned Laplacian, and normalized Laplacian settings.