StatisticalMechanics / CONCEPT

A regular biconnected graph with a positive fourth difference of ln(i! N(i))

Butera, Federbush and Pernici (arXiv:1502.06734) ask whether the bounds Delta^k ln(i! N(i)) <= 0 for k <= 4 always hold for regular biconnected graphs, where N(i) counts the configurations of i dimers. They do not: a 6-regular biconnected graph on 28 vertices, made of four copies of K_7 minus an edge joined in a ring, has a positive fourth difference at i = 10.