# Regular graph

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.[1] 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 disconnected 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 is strongly regular for any .

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

## Existence

It is well known that the necessary and sufficient conditions for a regular graph of order to exist are that and that is even. In such case it is easy to construct regular graphs by considering appropriate parameters for circulant graphs.

## Algebraic properties

Let A be the adjacency matrix of a graph. Then the graph is regular if and only if is an eigenvector of A.[2] Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to , so for such eigenvectors , we have .

A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one. The "only if" direction is a consequence of the Perron–Frobenius theorem.[2]

There is also a criterion for regular and connected graphs : a graph is connected and regular if and only if the matrix of ones J, with , is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).[3]

Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix . If G is not bipartite, then

[4]

## Generation

Regular graphs may be generated by the GenReg program.[5]

• Nash-Williams, Crispin (1969). "University of Waterloo Research Report". Waterloo, Ontario: University of Waterloo. |contribution= ignored (help)