Regular graph

Definition

Let $G$ be a graph 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 $k$. 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 is cubic. The complete graph $K_n$ is $(n-1)$-regular. By the handshaking lemma, 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.