{
  "id": "2008.05025",
  "version": "v1",
  "published": "2020-08-11T22:40:50.000Z",
  "updated": "2020-08-11T22:40:50.000Z",
  "title": "Constants and heat flow on graphs",
  "authors": [
    "Li Ma"
  ],
  "comment": "35 pages",
  "categories": [
    "math.AP",
    "math.CO",
    "math.DG",
    "math.SP"
  ],
  "abstract": "In this article, we first introduce the concepts of vector fields and their divergence, and we recall the concepts of the gradient, Laplacian operator, Cheeger constants, eigenvalues, and heat kernels on a locally finite graph $V$. We give a projective characteristic of the eigenvalues. We also give an extension of Barta Theorem. Then we introduce the mini-max value of a function on a locally finite and locally connected graph. We show that for a coercive function on on a locally finite and locally connected graph, there is a mini-max value of the function provided it has two strict local minima values. We consider the discrete Morse flow for the heat flow on a finite graph in the locally finite graph $V$. We show that under suitable assumptions on the graph one has a weak discrete Morse flow for the heat flow on $S$ on any time interval. We also study the heat flow with time-variable potential and its discrete Morse flow. We propose the concepts of harmonic maps from a graph to a Riemannian manifold and pose some open questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-11T22:40:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heat flow",
      "locally finite graph",
      "locally connected graph",
      "mini-max value",
      "strict local minima values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}