Consider a (possibly directed) graph $\mathcal{G}$ with nodes $\mathcal{V}$ and edges $\mathcal{E}$, and a positive target distribution $\pi$ over $\mathcal{V}$. Given a stochastic transition matrix $P$ over $\mathcal{V}$ such that:
(i) $P$ is irreducible and aperiodic, (ii) $P\pi=\pi$ and (iii) $P(i,j)=0$ if $(i,j)$ is not an edge of $\mathcal{G}$.
We can now define its conductance as
$\Phi(P) = \min_{\mathcal{X}\subset\mathcal{V};0<\pi(\mathcal{X})\leq\frac{1}{2}} \frac{\sum_{i\in \mathcal{X},j\notin\mathcal{X}}P_{j,i}\pi_i}{\pi(\mathcal{X})}.$
Is anything known about the maximum of $\Phi$ over all such $P$, i.e., $\max_{P:(i)-(iii)}\Phi(P)$?