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The occupancy fraction of a graph is a (normalized) measure on the size ofindependent sets under the hard-core model, depending on a variable (fugacity)$\lambda.$ We present a criterion for finding the graph with minimum occupancyfraction among graphs with a fixed order, and disprove five conjectures on theextremes of the occupancy fraction and (normalized) independence polynomial forcertain graph classes of regular graphs with a given girth.

Counterexamples to conjectures on the occupancy fraction of graphs