Sensitivity of mixing times, an example

We construct a family of growing finite bounded degree rooted graphs, $G_n$, in which the mixing time for simple random walk, starting at the root, is order $\log \mathrm{\mid }{G}_n\mathrm{\mid }$. Yet after a quasi - isometry, the ratio of $\mathrm{\mid }{G}_n\mathrm{\mid }$ over the mixing time grows arbitrarily slow to infinity.