Regular graph

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

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 2k + 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. The "only if" direction is a consequence of the Perron–Frobenius theorem.^[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 $J_{{ij}}=1$, 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 ${\displaystyle k=\lambda _{0}>\lambda _{1}\geq \cdots \geq \lambda _{n-1}}$. If G is not bipartite, then

${\displaystyle D\leq {\frac {\log {(n-1)}}{\log(\lambda _{0}/\lambda _{1})}}+1.}$^[4]

Generation

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

See also

• Random regular graph
• Strongly regular graph
• Moore graph
• Cage graph
• Highly irregular graph

References

1. Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientific. p. 29. ISBN 978-981-02-1859-1.
2. Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.
3. Curtin, Brian (2005), "Algebraic characterizations of graph regularity conditions", Designs, Codes and Cryptography, 34 (2-3): 241–248, doi:10.1007/s10623-004-4857-4, MR 2128333.
4. http://personal.plattsburgh.edu/quenelgt/pubpdf/diamest.pdf
5. 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". Waterloo, Ontario: University of Waterloo. |contribution= ignored (help)