Counterexamples to conjectures on the occupancy fraction of graphs

TL;DR

This paper investigates the occupancy fraction in graphs, providing a criterion for minimality and disproving five conjectures related to extremal properties in regular graphs with fixed girth.

Contribution

It introduces a new criterion for identifying graphs with minimal occupancy fraction and refutes several conjectures about extremal graph properties.

Findings

Disproved five conjectures on occupancy fraction extremes.

Provided a criterion for minimal occupancy fraction in fixed-order graphs.

Identified counterexamples in regular graphs with given girth.

Abstract

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.