  It is a fact simple to establish that the mixing time of the simple random
walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from
below by $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparing
the walk on $G_n$ to the walk on the infinite $d$-regular tree. If one can map
another infinite transitive graph onto $G_n$, then we can improve the strategy
by using a comparison with the random walk on this transitive graph (instead of
that of the regular tree), and we obtain a lower bound of the form $1/h \log
n$, where $h$ is the entropy rate associated with the walk on the transitive
graph. We call this the entropic lower bound. It was recently proved that in
the case of the tree, this entropic lower bound is sharp when graphs have
minimal spectral radius and thus that in that case the random walk exhibit
cutoff at the entropic time. In this paper, we provide a generalization of the
result by providing a sufficient condition on the spectra the random walks on
$G_n$ under which the random walk exhibit cutoff at the entropic time. It
applies notably to anisotropic random walks on random $d$-regular graphs and to
random walks on random $n$-lifts of a base graph (including non-reversible
walks).
