{
  "id": "2210.10966",
  "version": "v1",
  "published": "2022-10-20T02:25:18.000Z",
  "updated": "2022-10-20T02:25:18.000Z",
  "title": "Maximization of the first Laplace eigenvalue of a finite graph",
  "authors": [
    "T. Gomyou",
    "S. Nayatani"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "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 Goering-Helmberg-Wappler problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-20T02:25:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C62",
      "05C50"
    ],
    "keywords": [
      "first laplace eigenvalue",
      "finite graph",
      "first nonzero eigenvalue",
      "edge-length functions subject",
      "first-eigenvalue maximization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}