Threshold for uniqueness of minimizers of the occupancy fraction in G(n, α)

Establish the existence of a threshold value λ0 > 0 with the following property: for fixed integers n and α, among all n-vertex graphs G with independence number α, the minimizer of the occupancy fraction E_G(λ) = (1/n)·(λ P′_G(λ)/P_G(λ)) is unique for λ < λ0 and not unique for λ ≥ λ0; and determine the exact value of λ0, where P_G(λ) = ∑_{I∈𝒪(G)} λ^{|I|} is the independence polynomial of G.

Background

The paper studies the occupancy fraction E_G(λ) for graphs with fixed order n and independence number α, obtaining tight upper and lower bounds. For small λ (specifically 0 < λ < 2/(n−2)), the authors identify a specific extremal graph G1 = K_{n−α} ∨ αK1 that minimizes E_G(λ).

Based on computational evidence, the authors observe different behavior across λ, leading them to conjecture a transition point where the minimizer’s uniqueness changes. They explicitly state that they cannot currently determine this threshold’s exact value.