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arXiv · 2311.05542

Counterexamples to conjectures on the occupancy fraction of graphs

Stijn Cambie·Jorik Jooken
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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) $λ.$ 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.

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Stijn Cambie, Jorik Jooken. 2023-11-28. Counterexamples to conjectures on the occupancy fraction of graphs. https://arxiv.org/abs/2311.05542

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