Laplacian spectral characterization of some graphs obtained by product operation

A graph is said to be DLS, if there is no other non-isomorphic graph with the same Laplacian spectrum. Let GG be a DLS graph. We show that G×KrG×Kr is DLS if GG is disconnected. If GG is connected, it is proved that G×KrG×Kr is DLS under certain conditions. Applying this result, we prove that G×KrG×Kr is DLS if GG is a tree on n(n⩾5) vertices or a unicyclic graph on n(n>6) vertices.