In this paper we give a new method convert results dealing with graph theoretic (or Markov chain) Laplacians into results concerning Laplacians in analysis, such as on Riemannian manifolds. We illustrate this method by using the results of [6] prove $$ \λ 1\leq {1/over dist2(X,Y)} /Big(\cosh -1\sqrt {\mu Xc\mu Yc\over \mu X\mu Y}\Big)2. $$ for λ1 the first positive Neumann eigenvalue on a connected compact Riemannian manifold, and X, Y any two disjoint sets (and where Xc is the complement of X). This inequality has a version for the k-th positive eigenvalue (involving k + 1 disjoint sets), and holds more generally for all analytic Laplacians described in [6]. We show that this inequality is optimal to first order, in that it is impossible obtain an inequality of this form with the right-hand-side divided by 1 + ∊ for any fixed constant ∊ > 0.