Let G be a non-compact locally compact group with a continuous submultiplicative weight function ω such that ω(e)=1 and ω is diagonally bounded with bound K≥1. When G is σ-compact, we show that ⌊K⌋+1 many points in the spectrum of LUC(ω−1) are enough to determine the topological centre of LUC(ω−1)⁎ and that ⌊K⌋+2 many points in the spectrum of L∞(ω−1) are enough to determine the topological centre of L1(ω)⁎⁎ when G is in addition a SIN-group. We deduce that the topological centre of LUC(ω−1)⁎ is the weighted measure algebra M(ω) and that of C0(ω−1)⊥ is trivial for any locally compact group. The topological centre of L1(ω)⁎⁎ is L1(ω) and that of L0∞(ω)⊥ is trivial for any non-compact locally compact SIN-group. The same techniques apply and lead to similar results when G is a weakly cancellative right cancellative discrete semigroup.