  We performed numerical simulations of the $q$-state Potts model to compute
the reduced conductivity exponent $t/ \nu$ for the critical Coniglio-Klein
clusters in two dimensions, for values of $q$ in the range $[1;4]$. At
criticality, at least for $q<4$, the conductivity scales as $C(L) \sim
L^{-\frac{t}{\nu}}$, where $t$ and $\nu$ are, respectively, the conductivity
and correlation length exponents. For q=1, 2, 3, and 4, we followed two
independent procedures to estimate $t / \nu$. First, we computed directly the
conductivity at criticality and obtained $t / \nu$ from the size dependence.
Second, using the relation between conductivity and transport properties, we
obtained $t / \nu$ from the diffusion of a random walk on the backbone of the
cluster. From both methods, we estimated $t / \nu$ to be $0.986 \pm 0.012$,
$0.877 \pm 0.014$, $0.785 \pm 0.015$, and $0.658 \pm 0.030$, for q=1, 2, 3, and
4, respectively. We also evaluated $t /\nu$ for non integer values of $q$ and
propose the following conjecture $40gt/ \nu=72+20g-3g^2$ for the dependence of
the reduced conductivity exponent on $q$, in the range $ 0 \leq q \leq 4$,
where $g$ is the Coulomb gas coupling.
