  We investigate quantum lens spaces, $C(L_q^{2n+1}(r;\underline{m}))$,
introduced by Brzezi\'nski-Szyma\'nski as graph $C^*$-algebras. We give a new
description of $C(L_q^{2n+1}(r;\underline{m}))$ as graph $C^*$-algebras
amending an error in the original paper by Brzezi\'nski-Szyma\'nski.
Furthermore, for $n\leq 3$, we give a number-theoretic invariant, when all but
one weight are coprime to the order of the acting group $r$. This builds upon
the work of Eilers, Restorff, Ruiz and S{\o}rensen.
