Regular graph

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Graph families defined by their automorphisms
distance-transitive \boldsymbol{\rightarrow} distance-regular \boldsymbol{\leftarrow} strongly regular
symmetric (arc-transitive) \boldsymbol{\leftarrow} t-transitive, t ≥ 2 skew-symmetric
(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)
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.


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[edit]

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]


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

See also[edit]


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