Counting Independent Sets and Kernels of Regular Graphs

TL;DR

This paper provides numerical evidence supporting the conjecture that the number of independent sets and kernels in 3-regular graphs grow exponentially with small fluctuations, confirming the accuracy of the Bethe approximation.

Contribution

It offers the first numerical validation of the conjecture that fluctuations in the logarithm of independent sets and kernels are bounded for 3-regular graphs.

Findings

Average number of independent sets approaches w^n with w≈1.54563.

Number of kernels grows approximately as y^n with y≈1.299.

Fluctuations in the log of these counts are bounded by O(1).

Abstract

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).