Lower bound for strong nodal domains at the top eigenvalue of ternary Hamming graphs

Prove that every eigenfunction f of the Hamming graph H(n, 3) with Laplacian eigenvalue 3n has at least n + 1 strong nodal domains.

Background

For q ≥ 3, the paper studies the minimum number of strong nodal domains of eigenfunctions of H(n, q) with eigenvalue 3i when q = 3. It establishes the existence of eigenfunctions with exactly two strong nodal domains for every 1 ≤ i ≤ n − 1.

The case q = 3 and i = n is explicitly identified as the last remaining open case for q ≥ 3. Numerical experiments for n = 2, 3, and 4 found minimum strong-nodal-domain counts of 3, 4, and 5, respectively, motivating the conjectured lower bound n + 1.

References

In other words, the last remaining open case for q ≥ 3 is q = 3 and i = n. In our opinion, this case has something in common with the case q = 2, i = n − 1, and the minimum number of strong nodal domains seems to be a linear on n function. In particular, we conducted numerical experiments for n = 2, 3, 4, i = n and the functions with minimum number of strong nodal domains that we found have 3, 4 and 5 strong nodal domains respectively (see Figure 1). Based on these computations, we formulate the following conjecture. Conjecture 2. For any eigenfunction f of H(n, 3), n ≥ 1, with eigenvalue 3n we have SND(f ) ≥ n + 1.

— On strong nodal domains for eigenfunctions of Hamming graphs  (2502.14543 - Valyuzhenich et al., 20 Feb 2025) in Conjecture 2, Section 9, Concluding remarks, p. 10