On the number of domains of eigenfunctions in Hamming graphs.

Abstract:

For a given graph $G$, consider the eigenvector $\theta$ of its Laplacian. In this paper we focus on the so-called nodal domains (i.e. the connected components of the maximal induced subgraphs of $G$ on which an eigenvector $\theta$ does not change sign). In 2004 Bıyıkoğlu et al. started to study the question of lower and upper bounds on the number of domains in a binary hypercube (Hamming graph $H(n,2)$).

In this joint work with Alexandr Valyuzhenich (Hebei Normal University) we partially prove the conjecture from their work and continue the study for a wider class of graphs (Hamming graphs $H(n,q)$, $q\geq 2$).