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 internal vertex are equal to each other. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k.

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.

In analogy with the terminology for polynomials of low degrees, a 3-regular or 4-regular graph often is called a cubic graph or a quartic graph, respectively. Similarly, it is possible to denote k-regular graphs with ${\displaystyle k=5,6,7,8,\ldots }$ as quintic, sextic, septic, octic, et cetera.

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 is strongly regular for any m.

By the degree sum formula, a k-regular graph with n vertices has ${\displaystyle {\frac {nk}{2}}}$ edges. In particular, at least one of the order n and the degree k must be an even number.

A theorem by Nash-Williams says that every k‑regular graph on 2k + 1 vertices has a Hamiltonian cycle.

Let A be the adjacency matrix of a graph. Then the graph is regular if and only if ${\displaystyle {\textbf {j}}=(1,\dots ,1)}$ is an eigenvector of A. Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to ${\displaystyle {\textbf {j}}}$, so for such eigenvectors ${\displaystyle v=(v_{1},\dots ,v_{n})}$, we have ${\displaystyle \sum _{i=1}^{n}v_{i}=0}$.