  We consider arbitrary graphs $G$ with $n$ vertices and minimum degree at
least $\delta n$ where $\delta>0$ is constant. If the conductance of $G$ is
sufficiently large then we obtain an asymptotic expression for the cover time
$C_G$ of $G$ as the solution to an explicit transcendental equation. Failing
this, if the mixing time of a random walk on $G$ is of a lesser magnitude than
the cover time, then we can obtain an asymptotic deterministic estimate via a
decomposition into a bounded number of dense sub-graphs with high conductance.
Failing this we give a deterministic asymptotic (2+o(1))-approximation of
$C_G$.
