{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $\\{v_1, \\dots, v_n\\}$.\nThe adjacency matrix of $G$ is the $n \\times n$ matrix $A = (a_{ij})$ with\n$$a_{ij} = \\begin{cases} 1 \u0026 \\text{if } v_i \\sim v_j, \\\\ 0 \u0026 \\text{otherwise.} \\end{cases}$$\n\n## Remarks\n\nThe matrix $A$ is symmetric and has zeros on the diagonal. The sum of row $i$ is the [[degree|0T0F6CCS9W]] of $v_i$.\nThe entry $(A^k)_{ij}$ is the number of [[walks|0VJZM0F9YM]] of length $k$ from $v_i$ to $v_j$.\nAnother order of the vertices gives the matrix $P^{T} A P$ for a permutation matrix $P$.\n\n## Explanations\n\n### L1 human\n\nThe adjacency matrix is a table of zeros and ones. A $1$ in row $i$ and column $j$ means that vertices $i$ and $j$ are joined.\nFor a triangle, every entry off the diagonal is $1$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "08B9CX109G",
    "kind": "definition",
    "msc": [
      "05C50"
    ],
    "name": "Adjacency matrix",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Adjacency_matrix"
      }
    ],
    "tag": "0NZ8MDBN6Q",
    "uses": [
      "0JD5C82QMB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0NZ8MDBN6Q/",
  "links": [
    {
      "field": "defines",
      "target": "08B9CX109G"
    },
    {
      "field": "uses",
      "target": "0JD5C82QMB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0NZ8MDBN6Q.md",
  "msc": [
    "05C50"
  ],
  "name": "Adjacency matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0NZ/0NZ8MDBN6Q.md",
  "status": null,
  "tag": "0NZ8MDBN6Q",
  "uri": "m2://0NZ8MDBN6Q",
  "url": "https://math2.tamnd.com/wiki/adjacency-matrix-definition/",
  "writing": {
    "tag": "0NZ8MDBN6Q",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}