{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Summary\n\nIn spectral graph theory, the **Alon–Boppana bound** provides a lower bound on the second-largest eigenvalue of the [[adjacency matrix|08B9CX109G]] of a $d$-regular graph, meaning a graph in which every vertex has degree $d$. The reason for the interest in the second-largest eigenvalue is that the largest eigenvalue is guaranteed to be $d$ due to $d$-regularity, with the all-ones vector being the associated eigenvector. The graphs that come close to meeting this bound are Ramanujan graphs, which are examples of the best possible expander graphs.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "inventory": [
      "CQYSBR9K9P"
    ],
    "kind": "concept",
    "name": "Alon–Boppana bound",
    "same_as": {
      "wikidata": "Q85741005",
      "wikipedia": "Alon–Boppana bound"
    },
    "sources": [
      {
        "locator": "Alon–Boppana bound",
        "mention": "wikipedia://en/62480574",
        "note": "adopted",
        "sections": [
          "Summary"
        ],
        "wikipedia": "en:62480574@r1283895692"
      }
    ],
    "tag": "VQWGTT5P1R"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/VQWGTT5P1R/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/VQWGTT5P1R.md",
  "msc": null,
  "name": "Alon–Boppana bound",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VQW/VQWGTT5P1R.md",
  "status": null,
  "tag": "VQWGTT5P1R",
  "uri": "m2://VQWGTT5P1R",
  "url": "https://math2.tamnd.com/wiki/alon-boppana-bound/",
  "writing": {
    "tag": "VQWGTT5P1R",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge defines in",
        "note": "at least one definition",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of in",
        "note": "at least two examples",
        "have": 0,
        "want": 2
      },
      {
        "what": "edge not_instance_of in",
        "note": "at least one non-example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge specializes both",
        "note": "the more general notions and the special cases",
        "have": 0,
        "want": 1
      }
    ]
  }
}