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

TL;DR

This paper constructs a counterexample to a conjecture about monomer-dimer entropy in regular bipartite lattices, focusing on graphs with large C_4 and Theta_2,2,2 values, challenging previous assumptions.

Contribution

It introduces a novel lattice construction that disproves a prior conjecture relating to monomer-dimer entropy in bipartite graphs.

Findings

Counterexample to the conjecture on monomer-dimer entropy

Lattice with large C_4 and Theta_2,2,2 values

Challenges existing beliefs about regular bipartite lattices

Abstract

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