{
  "id": "1801.04555",
  "version": "v1",
  "published": "2018-01-14T13:34:30.000Z",
  "updated": "2018-01-14T13:34:30.000Z",
  "title": "Remarks on Graphons",
  "authors": [
    "Attila Nagy"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The notion of the graphon (a symmetric measurable fuzzy set of $[0, 1]^2$) was introduced by L. Lov\\'asz and B. Szegedy in 2006 to describe limit objects of convergent sequences of dense graphs. In their investigation the integral \\[t(F,W)=\\int _{[0, 1]^k}\\prod _{ij\\in E(F)}W(x_i,x_j)dx_1dx_2\\cdots dx_k\\] plays an important role in which $W$ is a graphon and $E(F)$ denotes the set of all edges of a $k$-labelled simple graph $F$. In our present paper we show that the set of all fuzzy sets of $[0, 1]^2$ is a right regular band with respect to the operation $\\circ$ defined by \\[(f\\circ g)(s,t)=\\vee _{(x,y)\\in [0, 1]^2}(f(x,y)\\wedge g(s,t));\\quad (s, t)\\in [0, 1]^2,\\] and the set of all graphons is a left ideal of this band. We prove that, if $W$ is an arbitrary graphon and $f$ is a fuzzy set of $[0, 1]^2$, then \\[|t(F; W)-t(F; f\\circ W)|\\leq |E(F)|(\\sup(W)-\\sup(f))\\Delta (\\{W> \\sup(f)\\} )\\] for arbitrary finite simple graphs $F$, where $\\Delta (\\{W> \\sup(f)\\})$ denotes the area of the set $\\{W>\\sup(f)\\}$ of all $(x, y)\\in [0, 1]^2$ satisfying $W(x,y)>\\sup(f)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-14T13:34:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "08A75",
      "20M10",
      "05C99"
    ],
    "keywords": [
      "arbitrary finite simple graphs",
      "symmetric measurable fuzzy set",
      "right regular band",
      "limit objects",
      "arbitrary graphon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}