# 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 2`k` + 1 vertices has a Hamiltonian cycle.

- 0-regular graph.svg
0-regular graph

- 1-regular graph.svg
1-regular graph

- 2-regular graph.svg
2-regular graph

- 3-regular graph.svg
3-regular graph

## 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.^{[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

## Generation

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

## See also

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑
^{2.0}^{2.1}Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998. - ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑
^{4.0}^{4.1}http://personal.plattsburgh.edu/quenelgt/pubpdf/diamest.pdf - ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}

## External links

- Weisstein, Eric W., "Regular Graph",
*MathWorld*. - Weisstein, Eric W., "Strongly Regular Graph",
*MathWorld*. - GenReg software and data by Markus Meringer.
- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}