---
tag: 0KA4S8R39M
kind: definition
name: Regular graph
msc: [05C07]
defines: 0Q70YFB6ZM
uses: [0T0F6CCS9W]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_graph
    locator: lead section
---

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $k$ be a natural number.
The graph $G$ is $k$-regular, or regular of degree $k$, if each vertex of $G$ has [[degree|0T0F6CCS9W]] $k$.
A graph is regular if it is $k$-regular for some $k$.

## Remarks

A $0$-regular graph has no edges. A $1$-regular graph is a perfect matching. A finite $2$-regular graph is a disjoint union of cycles.
A $3$-regular graph is also called cubic. The [[Petersen graph|0NR95CTFCZ]] is cubic.
The complete graph $K_n$ is $(n-1)$-regular. By the [[handshaking lemma|0XAP8WHNES]], a finite $k$-regular graph on $n$ vertices has $kn/2$ edges.

## Explanations

### L1 human

A graph is regular when all vertices have the same number of neighbours.
The corners and edges of a cube form a $3$-regular graph.