Lovasz number of random dense circulant graphs

Establish that for a random dense circulant graph G on n vertices, the expected Lovasz number satisfies E[ϑ(G)] = (1 + o(1)) √n.

Background

Random dense circulant graphs interpolate between fully random G(n,1/2) and deterministic Paley graphs. For circulant graphs, the Lovasz number admits equivalent linear program formulations via Fourier diagonalization.

Numerical evidence suggests that E ϑ(G) for random dense circulants matches the √n behavior conjectured for G(n,1/2). Proving this requires controlling LP solutions under Fourier-structured randomness, and partial progress gives an upper bound O(√(n log log n)).

References

We conjecture, based on numerical observations, that the Lov\ asz number for random dense circulant graphs behaves similarly to that for G(n, 1/2). This motivates the following question. Let G be distributed as a random dense circulant graph. Then, \begin{equation} \mathbb{E} \vartheta(G) = (1 + o(1))\sqrt{n}. \end{equation}

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Conjecture, Section “The Lovasz number of random circulant graphs (DD)” (Entry 9)