On strong nodal domains for eigenfunctions of Hamming graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909502984421376 |
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| author | Valyuzhenich, Alexandr Vorob'ev, Konstantin |
| author_facet | Valyuzhenich, Alexandr Vorob'ev, Konstantin |
| contents | The Laplacian matrix of the $n$-dimensional hypercube has $n+1$ distinct eigenvalues $2i$, where $0\leq i\leq n$. In 2004, Bıyıkoğlu, Hordijk, Leydold, Pisanski and Stadler initiated the study of eigenfunctions of hypercubes with the minimum number of weak and strong nodal domains. In particular, they proved that for every $1\leq i\leq \frac{n}{2}$ there is an eigenfunction of the hypercube with eigenvalue $2i$ that have exactly two strong nodal domains. Based on computational experiments, they conjectured that the result also holds for all $1\leq i\leq n-2$. In this work, we confirm their conjecture for $i\leq \frac{2}{3}(n-\frac{1}{2})$ if $i$ is odd and for $i\leq \frac{2}{3}(n-1)$ if $i$ is even. We also consider this problem for the Hamming graph $H(n,q)$, $q\geq 3$ (for $q=2$, this graph coincides with the $n$-dimensional hypercube), and obtain even stronger results for all $q\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14543 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On strong nodal domains for eigenfunctions of Hamming graphs Valyuzhenich, Alexandr Vorob'ev, Konstantin Combinatorics The Laplacian matrix of the $n$-dimensional hypercube has $n+1$ distinct eigenvalues $2i$, where $0\leq i\leq n$. In 2004, Bıyıkoğlu, Hordijk, Leydold, Pisanski and Stadler initiated the study of eigenfunctions of hypercubes with the minimum number of weak and strong nodal domains. In particular, they proved that for every $1\leq i\leq \frac{n}{2}$ there is an eigenfunction of the hypercube with eigenvalue $2i$ that have exactly two strong nodal domains. Based on computational experiments, they conjectured that the result also holds for all $1\leq i\leq n-2$. In this work, we confirm their conjecture for $i\leq \frac{2}{3}(n-\frac{1}{2})$ if $i$ is odd and for $i\leq \frac{2}{3}(n-1)$ if $i$ is even. We also consider this problem for the Hamming graph $H(n,q)$, $q\geq 3$ (for $q=2$, this graph coincides with the $n$-dimensional hypercube), and obtain even stronger results for all $q\geq 3$. |
| title | On strong nodal domains for eigenfunctions of Hamming graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.14543 |