Regular graph

Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive, t ≥ 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

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 ${\displaystyle k}$-regular graph of order ${\displaystyle n}$ to exist are that ${\displaystyle n\geq k+1}$ and that ${\displaystyle nk}$ 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 ${\displaystyle {\binom {n}{2}}={\dfrac {n(n-1)}{2}}}$ and degree here is ${\displaystyle n-1}$. So ${\displaystyle k=n-1,n=k+1}$. This is the minimum ${\displaystyle n}$ for a particular ${\displaystyle k}$. Also note that if any regular graph has order ${\displaystyle n}$ then number of edges are ${\displaystyle {\dfrac {nk}{2}}}$ so ${\displaystyle nk}$ 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 2k + 1 vertices has a Hamiltonian cycle.

Let A be the adjacency matrix of a graph. Then the graph is regular if and only if ${\displaystyle {\textbf {j}}=(1,\dots ,1)}$ is an eigenvector of A.^[2] Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to ${\displaystyle {\textbf {j}}}$, so for such eigenvectors ${\displaystyle v=(v_{1},\dots ,v_{n})}$, we have ${\displaystyle \sum _{i=1}^{n}v_{i}=0}$.

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 ${\displaystyle J_{ij}=1}$, 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

Fast algorithms exist to generate, up to isomorphism, all regular graphs with a given degree and number of vertices.^[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. pp. 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. Quenell, G. (1994-06-01). "Spectral Diameter Estimates for k-Regular Graphs". Advances in Mathematics. 106 (1): 122–148. doi:10.1006/aima.1994.1052. ISSN 0001-8708. Retrieved 2025-04-10.[1]
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.

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