{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G = (V, E)$ be a finite [[graph|0JD5C82QMB]] with $\\abs{V} \\ge 2$.\nFor $S \\subseteq V$, let $\\partial S$ be the set of edges with one end in $S$ and the other end in $V \\setminus S$.\nThe Cheeger constant of $G$ is\n$$h(G) = \\min \\set{ \\frac{\\abs{\\partial S}}{\\abs{S}} : S \\subseteq V, \\ 1 \\le \\abs{S} \\le \\frac{\\abs{V}}{2} }.$$\n\n## Remarks\n\nExample: in the complete graph $K_n$ with $n \\ge 2$, a set $S$ of size $k$ has $k(n - k)$ edges leaving it, so the ratio is $n - k$.\nThe minimum over $1 \\le k \\le n/2$ gives $h(K_n) = n - \\floor{n/2} = \\ceil{n/2}$.\nFor the path with $n$ vertices, the first $\\floor{n/2}$ vertices have one edge leaving them, so $h \\le 1/\\floor{n/2}$.\nThe bound $\\abs{S} \\le \\abs{V}/2$ cannot be dropped: for $S = V$ the ratio would be $0$ for every graph.\n$h(G) \u003e 0$ if and only if $G$ is [[connected|0FFAYNMAJZ]].\nIf $G$ is $d$-regular and $\\lambda_2$ is the second largest eigenvalue of its adjacency matrix, then $\\frac{d - \\lambda_2}{2} \\le h(G) \\le \\sqrt{2d(d - \\lambda_2)}$.\nThe notation $h(G)$ and the names edge expansion and isoperimetric number are also used.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "Y7S6W12N64",
    "kind": "definition",
    "msc": [
      "05C50",
      "05C40",
      "05C48"
    ],
    "name": "Cheeger constant",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cheeger_constant_(graph_theory)"
      }
    ],
    "tag": "GZKMX0WRA5",
    "uses": [
      "0JD5C82QMB",
      "0FFAYNMAJZ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/GZKMX0WRA5/",
  "links": [
    {
      "field": "defines",
      "target": "Y7S6W12N64"
    },
    {
      "field": "uses",
      "target": "0JD5C82QMB"
    },
    {
      "field": "uses",
      "target": "0FFAYNMAJZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GZKMX0WRA5.md",
  "msc": [
    "05C50",
    "05C40",
    "05C48"
  ],
  "name": "Cheeger constant",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GZK/GZKMX0WRA5.md",
  "status": null,
  "tag": "GZKMX0WRA5",
  "uri": "m2://GZKMX0WRA5",
  "url": "https://math2.tamnd.com/wiki/cheeger-constant-definition/",
  "writing": {
    "tag": "GZKMX0WRA5",
    "kind": "definition",
    "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
      }
    ]
  }
}