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