---
tag: C22WGRP3T8
kind: concept
name: Strongly regular graph
msc: ["05E30", 05C50]
specializes: [0Q70YFB6ZM]
sources:
  - url: https://en.wikipedia.org/wiki/Strongly_regular_graph
same_as:
  wikidata: Q692823
  wikipedia: Strongly regular graph
---

## Summary

A strongly regular graph is a [[regular graph|0Q70YFB6ZM]] with a second kind of regularity.
The number of common neighbours of two distinct vertices depends only on whether they are adjacent.
Its parameters are written $(v, k, \lambda, \mu)$.
A connected strongly regular graph that is not complete has exactly three distinct adjacency eigenvalues.
These graphs are a meeting point of graph theory, design theory and group theory.
The node [[strongly regular graph|J50MA9C0EH]] gives the precise definition.

## Explanations

### L1 human

In a strongly regular graph every vertex has the same number of neighbours.
Moreover, any two friends have the same number of common friends, and any two strangers have the same number of common friends.
The pentagon $C_5$ is an example: two adjacent vertices have no common neighbour, and two non-adjacent vertices have exactly one.
The [[Petersen graph|0NR95CTFCZ]] is a larger example.