Some graphs determined by their (signless) Laplacian spectra

A graph G is L–DS (respectively, Q–DS) if there is no other non-isomorphic graph with the same (respectively, signless) Laplacian spectrum as G  . Let G1∨G2G1∨G2 be the join graph of graphs G1G1 and G2G2, and Ur,n−rUr,n−r the graph obtained by attaching n−rn−r pendent vertices to a vertex of CrCr (the cycle of order r). In this paper, we prove that if G is L–DS and the algebraic connectivity of G   is less than three, then Kt∨GKt∨G is L–DS under certain condition, which extends the main result of Zhou and Bu (2012) [24]. Also, Ur,n−rUr,n−r is proved to be Q–DS   for r⩾3r⩾3.