正则图的独立集与核的计数
Counting Independent Sets and Kernels of Regular Graphs
摘要：
Chertkov，Gamarnik，Shah和Shin最近证明了对于大的n，大小为n且度为3的随机正则图的独立集的平均数接近于w^n，其中w约为1.54563，与Bethe近似一致。他们还提出了一个令人惊讶的猜想，即当n变大时，独立集个数的对数的波动只有O（1），这意味着贝特近似对所有3-正则图都是惊人的精确。在这里，我提供了支持这一猜想的数字证据，从使用二元决策图的独立集的精确计数。我还提供了支持新命题的数值证据，即大小为n的3-正则图的核数由y^n给出，其中y约为1.299，并且核数的对数的波动也仅为O（1）。
英文摘要：
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).
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