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 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 $K_{m}$ 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

Existence

It is well known that the necessary and sufficient conditions for a $k$ regular graph of order $n$ to exist are that $n\geq k+1$ and that $nk$ 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 ${\textbf {j}}=(1,\dots ,1)$ is an eigenvector of A.^[2] 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.^[2]

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

$D\leq {\frac {\log {(n-1)}}{\log(k/\lambda )}}+1$ ^[3]

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

Generation

Regular graphs may be generated by the GenReg program.^[4]

References

^ Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientific. p. 29. ISBN 978-981-02-1859-1.
^ ^a ^b Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.
^ http://personal.plattsburgh.edu/quenelgt/pubpdf/diamest.pdf
^ Meringer, Markus (1999). "Fast generation of regular graphs and construction of cages" (PDF). Journal of Graph Theory. 30 (2): 137–146. doi:10.1002/(SICI)1097-0118(199902)30:2<>1.0.CO;2-G.

External links

Weisstein, Eric W. "Regular Graph". MathWorld.
Weisstein, Eric W. "Strongly Regular Graph". MathWorld.
GenReg software and data by Markus Meringer.
Nash-Williams, Crispin (1969). "University of Waterloo Research Report" (Document). Waterloo, Ontario: University of Waterloo. {{cite document}}: Unknown parameter |contribution= ignored (help)

[1] Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientific. p. 29. ISBN 978-981-02-1859-1.

[Cvetkovic-2] Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.

[3] ttp://personal.plattsburgh.edu/quenelgt/pubpdf/diamest.pdf

[4] Meringer, Markus (1999). "Fast generation of regular graphs and construction of cages" (PDF). Journal of Graph Theory. 30 (2): 137–146. doi:10.1002/(SICI)1097-0118(199902)30:2<>1.0.CO;2-G.

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@@ Line 35: / Line 35: @@
 Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix <math>k=\lambda_0 >\lambda_1\geq \dots\geq\lambda_{n-1}</math>. If G is not bipartite
-<math>D\leq \frac{\log{(n-1)}}{\log(k/\lambda)}+1</math>
+<math>D\leq \frac{\log{(n-1)}}{\log(k/\lambda)}+1</math><ref>http://personal.plattsburgh.edu/quenelgt/pubpdf/diamest.pdf</ref>
 where <math> \lambda=\max_{i>0}\{\mid \lambda_i \mid \}</math>.{{citation needed|date=March 2009}}

Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive, $t \geq 2$		skew-symmetric
↓
_{(if connected)} vertex- and edge-transitive	→	edge-transitive and regular	→	edge-transitive
↓		↓		↓
vertex-transitive	→	regular	→	_{(if bipartite)} biregular
↑
Cayley graph	←	zero-symmetric		asymmetric

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