Most of the bounds I find in the literature are spectral (like
Cheeger inequalities and other bounds
on the algebraic connectivity). One could combine these bounds with
bounds on the spectrum itself, such as the one in this
article, to bound something like the sparsest cut, but this approach
provides no guarantees with respect to the actual width of
the cut (or maybe it does and it is just not clear to me?).
A (loosely) related upper bound is that the bisection width of
cubic graphs with $n$ vertices is at most $(frac16+epsilon)n,
epsilon>0$ (from this paper). Are there similar
worst-case bounds for heuristic bipartition algorithms?