{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "610F4NS6GD"
    }
  ],
  "body": "\n## Summary\n\nGraph enumeration is the part of combinatorics that counts [[graphs|00C0BR7D7H]] with a given property, either as labelled graphs or up to [[isomorphism|0F7QKGJS6B]].\nThere are $2^{\\binom{n}{2}}$ labelled graphs on $n$ given vertices, and [[Cayley's formula|0M2SSGSYPE]] counts the labelled trees.\nUnlabelled counts are harder, and tools such as the [[Pólya enumeration theorem|MPYQ39P01F]] and generating functions are used for them.\nIt is a central part of [[enumerative combinatorics|EX382NQTGP]].\n\n## Explanations\n\n### L1 human\n\nGraph enumeration asks \"how many graphs of this kind are there?\".\nThere are two ways to count.\nIn labelled counting the vertices have names, so two drawings with different names count as different graphs.\nIn unlabelled counting only the shape matters.\nFor three vertices there are $8$ labelled graphs but only $4$ shapes: no edge, one edge, a path with two edges, and a triangle.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "05C30",
      "05A16"
    ],
    "name": "Graph enumeration",
    "same_as": {
      "wikidata": "Q5597086",
      "wikipedia": "Graph enumeration"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Graph_enumeration"
      }
    ],
    "tag": "4AVF0RN7HV"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/4AVF0RN7HV/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/4AVF0RN7HV.md",
  "msc": [
    "05C30",
    "05A16"
  ],
  "name": "Graph enumeration",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4AV/4AVF0RN7HV.md",
  "status": null,
  "tag": "4AVF0RN7HV",
  "uri": "m2://4AVF0RN7HV",
  "url": "https://math2.tamnd.com/wiki/graph-enumeration/",
  "writing": {
    "tag": "4AVF0RN7HV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}