  Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive
graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to
taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$
converges in the Gromov Hausdorff distance to a torus of dimension $<q$,
equipped with some invariant Finsler metric. The proof relies on a recent
quantitative version of Gromov's theorem on groups with polynomial growth
obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and
$|X_n| = o\bigl({diam(X_n)^{\delta}}\bigr)$ for $\delta > 1$ sufficiently
small, we prove, this time by elementary means, that $(X_n)$ converges to a
circle.
