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 is called a graph or regular graph of degree .

Special cases

Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of cycless and infinite chains.

In analogy with the terminology for polynomials of low degrees, a or graph often is called a cubic graph or a quartic graph, respectively. Similarly, it is possible to denote k as quintic, sextic, septic, octic, etcetera.

A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every non-adjacent pair of vertices has the same number 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 .

Properties

By the degree sum formula, a -regular graph with edges. In particular, at least one of the order and the degree must be an even number.

A theorem by Nash-Williams says that every graph on vertices has a Hamiltonian cycle.

Let A be the adjacency matrix of a graph. Then the graph is regular if and only if is an eigenvector of A. Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues.

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.

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, is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).

Let G be a k-regular graph with diameter D. If G is not bipartite, then

Existence

-regular graph of order if and only if the natural numbers and is even.

Proof: If a graph with vertices is -regular, then the degree of any vertex v of vertices different from v, and indeed at least one of and must be even, whence so is their product.

Conversely, if and are two natural numbers satisfying both the inequality and the parity condition, then indeed there is a -regular circulant graph of order are adjacent). If in addition . Else is odd, whence .

, then this circulant graph is complete.

Generation

Fast algorithms exist to generate, up to isomorphism, all regular graphs with a given degree and number of vertices.

See also

• Random regular graph

• Strongly regular graph

• Moore graph

• Cage graph

• Highly irregular graph

References

External links

• GenReg software and data by Markus Meringer.