Regular graph: Difference between revisions

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

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 $\sum _{i=1}^{n}v_{i}=0$ .

When the graph is regular with degree k, the graph will be connected if and only if k has algebraic (and geometric) dimension 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 $J_{ij}=1$ , 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". MathWorld.
Weisstein, Eric W. "Strongly Regular Graph". MathWorld.
GenReg software 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

[Cvetkovic-1] Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.

[2] M. Meringer, J. Graph Theory, 1999, 30, 137.

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@@ Line 47: / Line 47: @@
 * {{MathWorld|urlname=RegularGraph|title=Regular Graph}}
 * {{MathWorld|urlname=StronglyRegularGraph|title=Strongly Regular Graph}}
+* [http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html GenReg] software by Markus Meringer.
 * {{Citation | last=Nash-Williams | first=Crispin |authorlink = Crispin St. J. A. Nash-Williams
 | contribution=Valency Sequences which force graphs to have Hamiltonian Circuits

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Langbahn Team – Weltmeisterschaft

Regular graph: Difference between revisions

Revision as of 22:41, 28 May 2009

Algebraic properties

Generation

See also

References

External links