Average kissing numbers for non-congruent sphere packings
  The Koebe circle packing theorem states that every finite planar graph can be realized as the nerve of a packing of (non-congruent) circles in R^3. We investigate the average kissing number of finite packings of non-congruent spheres in R^3 as a first restriction on the possible nerves of such packings. We show that the supremum k of the average kissing number for all packings satisfies   12.566 ~ 666/53 <= k < 8 + 4*sqrt(3) ~ 14.928   We obtain the upper bound by a resource exhaustion argument and the upper bound by a construction involving packings of spherical caps in S^3. Our result contradicts two naive conjectures about the average kissing number: That it is unbounded, or that it is supremized by an infinite packing of congruent spheres. 
Greg Kuperberg and Oded Schramm