Geodesic
Isoperimetric numbers of regular graphs of high degree with applications to arithmetic Riemann surfaces
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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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.
DOI : 10.37236/651
Classification : 05C75, 05C40, 30F10, 05C99, 53C20
Mots-clés : isoperimetric numbers, Platonic graphs, Cheeger constants 
@article{10_37236_651,
     author = {Dominic Lanphier and Jason Rosenhouse},
     title = {Isoperimetric numbers of regular graphs of high degree with applications to arithmetic {Riemann} surfaces},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/651},
     zbl = {1236.05167},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/651/}
}
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Dominic Lanphier; Jason Rosenhouse. Isoperimetric numbers of regular graphs of high degree with applications to arithmetic Riemann surfaces. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/651

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