{
  "id": "2208.01926",
  "version": "v1",
  "published": "2022-08-03T09:07:23.000Z",
  "updated": "2022-08-03T09:07:23.000Z",
  "title": "On adjacency operators of locally finite graphs",
  "authors": [
    "Vladimir I. Trofimov"
  ],
  "comment": "in Russian",
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $\\Gamma$ is called locally finite if, for each vertex $v$ of $\\Gamma$, the set $\\Gamma(v)$ of all neighbors of $v$ in $\\Gamma$ is finite. For any locally finite graph $\\Gamma$ with vertex set $V(\\Gamma)$ and for any field $F$, let $F^{V(\\Gamma)}$ be the vector space over $F$ of all functions $V(\\Gamma) \\to F$ (with natural componentwise operations) and let $A^{({\\rm alg})}_{\\Gamma,F}$ be the linear operator $F^{V(\\Gamma)} \\to F^{V(\\Gamma)}$ defined by $(A^{({\\rm alg})}_{\\Gamma,F}(f))(v) = \\sum_{u \\in \\Gamma(v)}f(u)$ for all $f \\in F^{V(\\Gamma)}$, $v \\in V(\\Gamma)$. In the case of finite graph $\\Gamma$ the mapping $A^{({\\rm alg})}_{\\Gamma,F}$ is the well known operator defined by the adjacency matrix of $\\Gamma$ (over $F$), and the theory of eigenvalues and eigenfunctions of such operator is a well-developed (at least in the case $F = \\mathbb{C}$) part of the theory of finite graphs. In this paper we develope a theory of eigenvalues and eigenfunctions of $A^{({\\rm alg})}_{\\Gamma,F}$ for arbitrary infinite locally finite graphs $\\Gamma$ (although a few results may be of interest for finite graphs) and fields $F$ with a special emphasis on the case when $\\Gamma$ is connected with uniformly bounded vertex degrees and $F = \\mathbb{C}$. By the author opinion, previous attempts in this direction were not quite satisfactory since were limited by consideration of rather special eigenfunctions and corresponding eigenvalues.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-03T09:07:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C63",
      "05C50"
    ],
    "keywords": [
      "adjacency operators",
      "arbitrary infinite locally finite graphs",
      "eigenfunctions",
      "eigenvalues",
      "adjacency matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "ru",
      "license": "arXiv",
      "status": "editable"
    }
  }
}