Two strong nodal domains for hypercube eigenfunctions in the full conjectured range

Establish that for every integer n and every index i satisfying 1 ≤ i ≤ n − 2, the n-dimensional hypercube H(n, 2) has an eigenfunction with Laplacian eigenvalue 2i and exactly two strong nodal domains.

Background

Bıyıkoğlu, Hordijk, Leydold, Pisanski, and Stadler proved that for 1 ≤ i ≤ n/2 there exists an eigenfunction of H(n, 2) with eigenvalue 2i and exactly two strong nodal domains. They formulated a broader conjecture, based on computational experiments, extending this conclusion to every 1 ≤ i ≤ n − 2.

The paper proves the conjecture only for i up to approximately 2n/3, with the precise bound depending on the parity of i. Consequently, the conjectured assertion remains unresolved for the remaining indices in the stated range.

References

Based on computational experiments, Bıyıkoğlu et al. [4] conjectured that Theorem 3 can be extended as follows. Conjecture 1 ([4], Conjecture 1). For every 1 ≤ i ≤ n − 2, there is an eigenfunction f of H(n, 2) with eigenvalue 2i such that SND(f ) = 2.

— On strong nodal domains for eigenfunctions of Hamming graphs  (2502.14543 - Valyuzhenich et al., 20 Feb 2025) in Conjecture 1, Section 1, Introduction, p. 3