Regular graph

---

title: "Regular graph" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["graph-families", "regular-graphs"] description: "Graph where each vertex has the same number of neighbors" topic_path: "general/graph-families" source: "https://en.wikipedia.org/wiki/Regular_graph" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Graph where each vertex has the same number of neighbors ::

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. 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.

In analogy with the terminology for polynomials of low degrees, a 3-regular or 4-regular graph often is called a cubic graph or a quartic graph, respectively. Similarly, it is possible to denote k-regular graphs with k=5,6,7,8,\ldots as quintic, sextic, septic, octic, et cetera.

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 is strongly regular for any m.

Image:0-regular_graph.svg|0-regular graph Image:1-regular_graph.svg|1-regular graph Image:2-regular_graph.svg|2-regular graph Image:3-regular_graph.svg|3-regular graph

Properties

By the degree sum formula, a k-regular graph with n vertices has \frac{nk}2 edges. In particular, at least one of the order n and the degree k must be an even number.

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 \textbf{j}=(1, \dots ,1) is an eigenvector of A. 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.

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).{{citation | last = Curtin | first = Brian | doi = 10.1007/s10623-004-4857-4 | issue = 2–3 | journal = Designs, Codes and Cryptography | mr = 2128333 | pages = 241–248 | title = Algebraic characterizations of graph regularity conditions | volume = 34 | year = 2005}}.

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.

Existence

There exists a k-regular graph of order n if and only if the natural numbers n and k satisfy the inequality n \geq k+1 and that nk is even.

Proof: If a graph with n vertices is k-regular, then the degree k of any vertex v cannot exceed the number n-1 of vertices different from v, and indeed at least one of n and k must be even, whence so is their product.

Conversely, if n and k are two natural numbers satisfying both the inequality and the parity condition, then indeed there is a k-regular circulant graph C_n^{s_1,\ldots,s_r} of order n (where the s_i denote the minimal jumps' such that vertices with indices differing by an s_i are adjacent). If in addition k is even, then k = 2r, and a possible choice is (s_1,\ldots,s_r) = (1,2,\ldots,r). Else k is odd, whence n must be even, say with n = 2m, and then k = 2r-1 and the jumps' may be chosen as (s_1,\ldots,s_r) = (1,2,\ldots,r-1,m).

If n=k+1, then this circulant graph is complete.

Generation

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

References

References

1. Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.
2. Quenell, G.. (1994-06-01). "Spectral Diameter Estimates for k-Regular Graphs". Advances in Mathematics.
3. Meringer, Markus. (1999). "Fast generation of regular graphs and construction of cages". [[Journal of Graph Theory]].

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::