Regular graph: Definition from Answers.com

Graph families defined by their automorphisms
distance-transitive	$\rightarrow$	distance-regular	$\leftarrow$	strongly regular
$\downarrow$
symmetric (arc-transitive)	$\leftarrow$	t-transitive, t ≥ 2
$\downarrow$ _{(if connected)}
vertex- and edge-transitive	$\rightarrow$	edge-transitive and regular	$\rightarrow$	edge-transitive
$\downarrow$		$\downarrow$
vertex-transitive	$\rightarrow$	regular
$\uparrow$
Cayley graph				skew-symmetric

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 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.

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 $K m$ is strongly regular for any $m$ .

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

0-regular graph

1-regular graph

2-regular graph

3-regular graph

1 Algebraic properties
2 Generation
3 See also
4 References
5 External links

Algebraic properties

Let A be the adjacency matrix of a graph. Then the graph is regular if and only if $\textbf{j}=(1, \dots ,1)$ is an eigenvector of A.^[1] Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to $\textbf{j}$ , so for such eigenvectors $v=(v_1,\dots,v_n)$ , we have $\sum_{i=1}^n v_i = 0$ .

A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one.^[1]

There is also a criterion for regular and connected graphs : a graph is connected and regular if and only if the matrix J, with $J i j = 1$ , is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).^{[citation needed]}

Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix $k=\lambda_0 >\lambda_1\geq \dots\geq\lambda_{n-1}$ . If G is not bipartite

$D\leq \frac{\log{(n-1)}}{\log(k/\lambda)}+1$

where $\lambda=\max_{i>0}\{\mid \lambda_i \mid \}$ .^{[citation needed]}

Generation

Regular graphs may be generated by GenReg program. ^[2]

References

^ ^a ^b Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.
^ M. Meringer, J. Graph Theory, 1999, 30, 137.

External links

Weisstein, Eric W., "Regular Graph" from MathWorld.
Weisstein, Eric W., "Strongly Regular Graph" from MathWorld.
GenReg software and data by Markus Meringer.
Nash-Williams, Crispin (1969), "Valency Sequences which force graphs to have Hamiltonian Circuits", University of Waterloo Research Report, Waterloo, Ontario: University of Waterloo

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Regular graph

Contents

Algebraic properties

Generation

See also

References

External links