Maximization of the first Laplace eigenvalue of a finite graph
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914967217766400 |
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| author | Gomyou, T. Nayatani, S. |
| author_facet | Gomyou, T. Nayatani, S. |
| contents | Given a length function on the edge set of a finite graph, we define a vertex-weight and an edge-weight in terms of it and consider the corresponding graph Laplacian. In this paper, we consider the problem of maximizing the first nonzero eigenvalue of this Laplacian over all edge-length functions subject to a certain normalization. For an extremal solution of this problem, we prove that there exists a map from the vertex set to a Euclidean space consisting of first eigenfunctions of the corresponding Laplacian so that the length function can be explicitly expressed in terms of the map and the Euclidean distance. This is a graph-analogue of Nadirashvili's result related to first-eigenvalue maximization problem on a smooth surface. We discuss simple examples and also prove a similar result for a maximizing solution of the Göring-Helmberg-Wappler problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_10966 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Maximization of the first Laplace eigenvalue of a finite graph Gomyou, T. Nayatani, S. Combinatorics 05C62 (Primary) 05C50 (Secondary) Given a length function on the edge set of a finite graph, we define a vertex-weight and an edge-weight in terms of it and consider the corresponding graph Laplacian. In this paper, we consider the problem of maximizing the first nonzero eigenvalue of this Laplacian over all edge-length functions subject to a certain normalization. For an extremal solution of this problem, we prove that there exists a map from the vertex set to a Euclidean space consisting of first eigenfunctions of the corresponding Laplacian so that the length function can be explicitly expressed in terms of the map and the Euclidean distance. This is a graph-analogue of Nadirashvili's result related to first-eigenvalue maximization problem on a smooth surface. We discuss simple examples and also prove a similar result for a maximizing solution of the Göring-Helmberg-Wappler problem. |
| title | Maximization of the first Laplace eigenvalue of a finite graph |
| topic | Combinatorics 05C62 (Primary) 05C50 (Secondary) |
| url | https://arxiv.org/abs/2210.10966 |