Average Nodal Count and the Nodal Count Condition for Graphs
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arXiv
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| Format: | Preprint |
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2024
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| author | Alon, Lior Urschel, John |
| author_facet | Alon, Lior Urschel, John |
| contents | The nodal edge count of an eigenvector of the Laplacian of a graph is the number of edges on which it changes sign. This quantity extends to any real symmetric $n\times n$ matrix supported on a graph $G$ with $n$ vertices. The average nodal count, averaged over all eigenvectors of a given matrix, is known to be bounded between $\frac{n-1}{2}$ and $\frac{n-1}{2}+β(G)$, where $β(G)$ is the first Betti number of $G$ (a topological quantity), and it was believed that generically the average should be around $\frac{n-1}{2}+β(G)/2$. We prove that this is not the case: the average is bounded between $\frac{n-1}{2}+β(G)/n$ and $\frac{n-1}{2}+β(G)-β(G)/n$, and we provide graphs and matrices that attain the upper and lower bounds for any possible choice of $n$ and $β$.
A natural condition on a matrix for defining the nodal count is that it has simple eigenvalues and non-vanishing eigenvectors. For any connected graph $G$, a generic real symmetric matrix supported on $G$ satisfies this nodal count condition. However, the situation for constant diagonal matrices is far more subtle. We completely characterize the graphs $G$ for which this condition is generically true, and show that if this is not the case, then any real symmetric matrix supported on $G$ with constant diagonal has a multiple eigenvalue or an eigenvector that vanishes somewhere. Finally, we discuss what can be said when this nodal count condition fails, and provide examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03151 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Average Nodal Count and the Nodal Count Condition for Graphs Alon, Lior Urschel, John Mathematical Physics Combinatorics 05C50 (Primary) 15A42, 15B35 (Secondary) The nodal edge count of an eigenvector of the Laplacian of a graph is the number of edges on which it changes sign. This quantity extends to any real symmetric $n\times n$ matrix supported on a graph $G$ with $n$ vertices. The average nodal count, averaged over all eigenvectors of a given matrix, is known to be bounded between $\frac{n-1}{2}$ and $\frac{n-1}{2}+β(G)$, where $β(G)$ is the first Betti number of $G$ (a topological quantity), and it was believed that generically the average should be around $\frac{n-1}{2}+β(G)/2$. We prove that this is not the case: the average is bounded between $\frac{n-1}{2}+β(G)/n$ and $\frac{n-1}{2}+β(G)-β(G)/n$, and we provide graphs and matrices that attain the upper and lower bounds for any possible choice of $n$ and $β$. A natural condition on a matrix for defining the nodal count is that it has simple eigenvalues and non-vanishing eigenvectors. For any connected graph $G$, a generic real symmetric matrix supported on $G$ satisfies this nodal count condition. However, the situation for constant diagonal matrices is far more subtle. We completely characterize the graphs $G$ for which this condition is generically true, and show that if this is not the case, then any real symmetric matrix supported on $G$ with constant diagonal has a multiple eigenvalue or an eigenvector that vanishes somewhere. Finally, we discuss what can be said when this nodal count condition fails, and provide examples. |
| title | Average Nodal Count and the Nodal Count Condition for Graphs |
| topic | Mathematical Physics Combinatorics 05C50 (Primary) 15A42, 15B35 (Secondary) |
| url | https://arxiv.org/abs/2404.03151 |