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{\bf Dominic Lanphier and Jason Rosenhouse}
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{\bf Isoperimetric Numbers of Regular Graphs of High Degree with Applications to Arithmetic Riemann Surfaces}
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We derive upper and lower bounds on the isoperimetric numbers and bisection widths of a large class of regular graphs of high degree.
Our methods are combinatorial and do not require a knowledge of the eigenvalue spectrum.
We apply these bounds to random regular graphs of high degree and the Platonic graphs over the rings $\mathbb{Z}_n$. In the latter case we show that these graphs are generally non-Ramanujan for composite $n$ and we also give sharp asymptotic bounds for the isoperimetric numbers. We conclude by giving bounds on the Cheeger constants
of arithmetic Riemann surfaces. For a large class of these surfaces these bounds are an improvement over the known asymptotic bounds.
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