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 internal 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 $k$ .

Special cases

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 a disjoint union of 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 $m$ .

0-regular graph
1-regular graph
2-regular graph
3-regular graph

Existence

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.

Proof: A complete graph has every pair of distinct vertices connected to each other by a unique edge. So edges are maximum in complete graph and number of edges are ${\binom {n}{2}}={\dfrac {n(n-1)}{2}}$ and degree here is $n-1$ . So $k=n-1,n=k+1$ . This is the minimum $n$ for a particular $k$ . Also note that if any regular graph has order $n$ then number of edges are ${\dfrac {nk}{2}}$ so $nk$ has to be even. In such case it is easy to construct regular graphs by considering appropriate parameters for circulant graphs.

Properties

From the handshaking lemma, a $k$ -regular graph with odd $k$ has an even number of vertices.

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

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 $k=\lambda _{0}>\lambda _{1}\geq \cdots \geq \lambda _{n-1}$ . If G is not bipartite, then

D\leq {\frac {\log {(n-1)}}{\log(\lambda _{0}/\lambda _{1})}}+1.

^[4]

Generation

Fast algorithms exist to generate, up to isomorphism, all regular graphs with a given degree and number of vertices.^[5]

References

^ Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientific. pp. 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.
^ 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.
^ [1]^{[citation needed]}
^ 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<137::AID-JGT7>3.0.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), Valency Sequences which force graphs to have Hamiltonian Circuits, University of Waterloo Research Report, Waterloo, Ontario: University of Waterloo

[1] Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientific. pp. 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] 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] [1]^{[citation needed]}

[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<137::AID-JGT7>3.0.CO;2-G.

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[5]

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