  Chandrasekaran, Chertkov, Gamarnik, Shah, and Shin recently proved that the
average number of independent sets of random regular graphs of size n and
degree 3 approaches w^n for large n, where w is approximately 1.54563,
consistent with the Bethe approximation. They also made the surprising
conjecture that the fluctuations of the logarithm of the number of independent
sets were only O(1) as n grew large, which would mean that the Bethe
approximation is amazingly accurate for all 3-regular graphs. Here, I provide
numerical evidence supporting this conjecture obtained from exact counts of
independent sets using binary decision diagrams. I also provide numerical
evidence that supports the novel conjectures that the number of kernels of
3-regular graphs of size n is given by y^n, where y is approximately 1.299, and
that the fluctuations in the logarithm of the number of kernels is also only
O(1).
