# Extension of Cheeger's inequality with distinguished vertices

The standard Cheeger's inequality for graph $G$ states that

$\frac{1}{2}$ $\lambda$ < $\phi(G)$ < $\sqrt{2\lambda}$

where $\lambda$ is the second smallest eigenvalue of the normalized Laplacian of G, and $\phi(G)$ is the conductance of G.

Now suppose we are also given two special vertices $s$ and $t$, and would like to consider only cuts that separate $s$ and $t$. Can we have a similar result relating the conductance in this special case, and some Rayleigh coefficient of the normalized Laplacian of the graph?

-