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The quantities C_4 and Theta_2,2,2 are as defined by Wanless, C_4 just thenumber of 4-loops of a graph. The construction of this paper provides acounterexample to a conjecture of Butera, Pernici, and the author about themonomer-dimer entropy, lambda, of a regular bipartite lattice. The lattice weconstruct is not a lattice graph in its most common definition.

Regular Bipartite Lattices with Large Values of Theta_2,2,2/C_4