  Extending the work of Alon, Frieze abnd Welsh, we show that there are
randomized polynomial time approximation schemes for computing the Tutte
polynomial in subdense graphs with an minimal node degree of $\Omega\left (
\frac{n}{\sqrt{\log n}}\right )$ . The same holds for the partition function
$Z$ in the random cluster model with uniform edge probabilities and for the
associated distribution $\lambda (A),\: A \subseteq E$ whenever the underlying
graph $G=(V,E)$ is $c\cdot\frac{n}{\sqrt{\log (n)}}$-subdense. In the
superdense case with node degrees $n-o(n)$, we show that the Tutte polynomial
$T_G(x,y)$ is asymptotically equal to $Q=(x-1)(y-1)$. Moreover, we briefly
discuss the problem of approximating $Z$ in the case of $(\alpha, \beta
)$-power law graphs.
