Eigenvalue ratios of nonnegatively curved graphs

We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality $CD(0,\infty)$. This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs satisfying $CD(0,\infty)$. We also discuss a higher order Cheeger constant ratio estimate and related topics about expanders.