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 G be a DLS graph. We show that G × K r is DLS if G is disconnected. If G is connected, it is proved that G × K r is DLS under certain conditions. Applying this result, we prove that G × K r is DLS if G is a tree on n ( n ⩾ 5 ) vertices or a unicyclic graph on n ( n > 6 ) vertices.