Regular graph

Template:Graph families defined by their automorphisms 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.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 $K m$ is strongly regular for any $m$ .

0-regular graph

0-regular graph
1-regular graph

1-regular graph
2-regular graph

2-regular graph
3-regular graph

3-regular graph

Properties

By the degree sum formula, a $k$ -regular graph with $n$ vertices has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 $2 k + 1$ vertices has a Hamiltonian cycle.

Let A be the adjacency matrix of a graph. Then the graph is regular if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{j}} , so for such eigenvectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=(v_1,\dots,v_n)} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\lambda_0 >\lambda_1\geq \cdots\geq\lambda_{n-1}} . If G is not bipartite, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D\leq \frac{\log{(n-1)}}{\log(\lambda_0/\lambda_1)}+1. } ^[4]

Existence

There exists a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} -regular graph of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} if and only if the natural numbers $n$ and $k$ satisfy the inequality Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq k+1 } and that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_n^{s_1,\ldots,s_r}} of order $n$ (where the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} denote the minimal `jumps' such that vertices with indices differing by an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} are adjacent). If in addition $k$ is even, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = 2r} , and a possible choice is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s_1,\ldots,s_r) = (1,2,\ldots,r)} . Else $k$ is odd, whence $n$ must be even, say with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 2m} , and then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = 2r-1} and the `jumps' may be chosen as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s_1,\ldots,s_r) = (1,2,\ldots,r-1,m)} .

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.^[5]

References

↑ Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientific. pp. 29. ISBN 978-981-02-1859-1.
↑ ^2.0 ^2.1 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.
↑ 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]
↑ 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] 2.0 ^2.1 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.

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Regular graph

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Existence

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Existence

Generation

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