---
tag: VQWGTT5P1R
kind: concept
name: Alon–Boppana bound
sources:
  - wikipedia: en:62480574@r1283895692
    locator: Alon–Boppana bound
    note: adopted
    mention: wikipedia://en/62480574
    sections: [Summary]
same_as:
  wikidata: Q85741005
  wikipedia: Alon–Boppana bound
inventory: [CQYSBR9K9P]
---

## Summary

In 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.