Bounding ${\lambda }_2$ for K\"ahler-Einstein metrics with large symmetry groups

We calculate an upper bound for the second nonzero eigenvalue of the scalar Laplacian, ${\lambda }_2$, for toric K"ahler-Einstein metrics in terms of the polytope data. We give some detailed examples in complex dimensions 1, 2 and 3. We also discuss extensions of this method to other geometries.