Ed wrote: << I found this paper about the Klein graph quite nice. http://www.liga.ens.fr/~deza/withFowler/AdditionKlein.pdf They call it the Plumber's Nightmare Graph, which is an interesting name.

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In case anyone is interested in some context for this: This is the edge graph (the 1-skeleton) of the so-called "regular map" of the 3-holed torus by 24 heptagons, 3 per vertex. A regular map on a surface has a combinatorial automorphism group that's transitive on flags: A flag is any triple given by (v,e,f) for v = vertex lying in e = edge lying in f = face. There are 7x24/3 = 56 vertices, each lying in 3 edges, each lying in 2 faces, giving a total of 56x3x2 = 336 flags, and by transitivity 336 is also the order of the automorphism group. What's special about this particular 3-holed torus (as a Riemann surface) is that for any compact Riemann surface of genus >= 2, its group of conformal transformations G is finite and must satify Hurwitz's bound #(G) <= 84(g-1). There is a *unique* Riemann surface M_3 of genus 3 (up to conformal equivalence) that reaches this bound of 84x2 = 168 conformal equivalences, and since no Riemann surface of genus 2 meets Hurwitz's bound, this is the "first" Riemann surface that achieves this maximal symmetry. The group Aut(M_3) of order 168 is the unique simple group of order 168, mysteriously isomorphic to both PSL(3,Z_2) and PSL(2,Z_7). Is is also a subgroup of the Lie group PSL(2,C). If anti-conformal equivalences are thrown in as well, one gets the same group G of order 336 above. The Riemann surface M_3 has many (real analytic) metric realizations, among them one of constant negative curvature (unique if the curvature is fixed, at say -1). This can be constructed from 24 regular heptagons in the hyperbolic plane, each of the unique size that allows exactly 3 of them to fit about any vertex. But it has an even more beautiful metric realization M_K given by one homogeneous complex polynomial defined on the complex projective plane CP^2: the polynomial was discovered by Klein and is X Y^3 + Y Z^3 + Z X^3 = 0 (where CP^2 is defined as the quotient of C^3 - {0} by (X Y Z) ~ (cX cY cZ) for all nonzero c in C). M_K does not have constant curvature, but it is a minimal surface in CP^2. --Dan