Let G G be a locally compact SIN group, and w w a weight on G G which is diagonally bounded with bound K K on a dispersed set. We show that the topological centre of L U C ( w − 1 ) ∗ {LUC}(w^{-1})^* equals M ( w ) M(w) , and there are ⌊ K ⌋ + 1 \lfloor K \rfloor + 1 many points in the spectrum of L U C ( w − 1 ) {LUC}(w^{-1}) which are determining for the topological centre (DTC). As corollaries, we obtain DTC results as well for L 1 ( w ) ∗ ∗ L_1(w)^{**} , C 0 ( w − 1 ) ⊥ C_0(w^{-1})^\perp and L ∞ , 0 ( w − 1 ) ⊥ L_{\infty , 0}(w^{-1})^\perp . We also give a short proof of the first result in the unweighted case for all locally compact second countable abelian groups G G .