{
  "id": "1401.4623",
  "version": "v2",
  "published": "2014-01-19T00:47:21.000Z",
  "updated": "2014-12-09T12:08:50.000Z",
  "title": "The magnitude of a graph",
  "authors": [
    "Tom Leinster"
  ],
  "comment": "19 pages. Version 2: introduction rewritten; substance unchanged",
  "categories": [
    "math.CO",
    "math.CT"
  ],
  "abstract": "The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its cardinality-like properties are multiplicativity with respect to cartesian product and an inclusion-exclusion formula for the magnitude of a union. Formally, the magnitude of a graph is both a rational function over Q and a power series over Z. It shares features with one of the most important of all graph invariants, the Tutte polynomial; for instance, magnitude is invariant under Whitney twists when the points of identification are adjacent. Nevertheless, the magnitude of a graph is not determined by its Tutte polynomial, nor even by its cycle matroid, and it therefore carries information that they do not.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-19T00:47:21.000Z",
      "abstract": "The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and, conjecturally, geometric measure of several kinds. Among its cardinality-like properties are multiplicativity with respect to cartesian product and an inclusion-exclusion formula for the magnitude of a union. Formally, the magnitude of a graph is both a rational function over Q and a power series over Z. It resembles the Tutte polynomial in some respects: for instance, the magnitude of a tree depends only on its order, and magnitude is invariant under Whitney twists when the points of identification are adjacent. Nevertheless, the magnitude of a graph is not determined by its Tutte polynomial, nor even by its cycle matroid, and it therefore contains information that they do not.",
      "comment": "19 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-09T12:08:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C31",
      "05C12",
      "18D20"
    ],
    "keywords": [
      "tutte polynomial",
      "cartesian product",
      "inclusion-exclusion formula",
      "rational function",
      "contains information"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.4623L"
    }
  }
}