Let $G=(V,E)$ be a graph. The gradient of a function $f:V\longrightarrow R$ is defined on the edges of the graph, given by the discrete derivative $ \nabla f(e)=f(y)-f(x), \quad e=(x,y) \in E $
Take $V=S_{2n}$ the group of permutations $\pi$ of the set $\{1, \ldots 2n\}$ and $e=(\pi, \pi \tau) \in E$ for any transposition $\tau$.
I am trying to find a lower bound of the spectral gap of the graph $(S_{2n}, E)$. I guess it should be well-known. Any literature would be very helpful.
Thank you.