In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other. A regular graph with vertices of degree
k
{\displaystyle k}
is called a
k
{\displaystyle k}
‑regular graph or regular graph of degree
k
{\displaystyle k}
. Also, from the handshaking lemma, a regular graph contains an even number of vertices with odd degree.
Regular graphs of degree at most 2 are easy to classify: a 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of a disjoint union of cycles and infinite chains.
A 3-regular graph is known as a cubic graph.
A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common. The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices.
The complete graph
K
m
{\displaystyle K_{m}}
is strongly regular for any
m
{\displaystyle m}
.
A theorem by Nash-Williams says that every
k
{\displaystyle k}
‑regular graph on 2k + 1 vertices has a Hamiltonian cycle.