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.
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 Km is strongly regular for any m.
Existence
The necessary and sufficient conditions for a k {\displaystyle k} -regular graph of order n {\displaystyle n} to exist are that n ≥ k + 1 {\displaystyle n\geq k+1} and that n k {\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 ( n 2 ) = n ( n − 1 ) 2 {\displaystyle {\binom {n}{2}}={\dfrac {n(n-1)}{2}}} and degree here is n − 1 {\displaystyle n-1} . So k = n − 1 , n = k + 1 {\displaystyle k=n-1,n=k+1} . This is the minimum n {\displaystyle n} for a particular k {\displaystyle k} . Also note that if any regular graph has order n {\displaystyle n} then number of edges are n k 2 {\displaystyle {\dfrac {nk}{2}}} so n k {\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 j = ( 1 , … , 1 ) {\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 j {\displaystyle {\textbf {j}}} , so for such eigenvectors v = ( v 1 , … , v n ) {\displaystyle v=(v_{1},\dots ,v_{n})} , we have ∑ i = 1 n v i = 0 {\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.3
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 i j = 1 {\displaystyle J_{ij}=1} , is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).4
Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix k = λ 0 > λ 1 ≥ ⋯ ≥ λ n − 1 {\displaystyle k=\lambda _{0}>\lambda _{1}\geq \cdots \geq \lambda _{n-1}} . If G is not bipartite, then
D ≤ log ( n − 1 ) log ( λ 0 / λ 1 ) + 1. {\displaystyle D\leq {\frac {\log {(n-1)}}{\log(\lambda _{0}/\lambda _{1})}}+1.} 5Generation
Fast algorithms exist to generate, up to isomorphism, all regular graphs with a given degree and number of vertices.6
See also
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
References
Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientific. pp. 29. ISBN 978-981-02-1859-1. 978-981-02-1859-1 ↩
Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998. ↩
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. /wiki/Doi_(identifier) ↩
[1][citation needed] http://personal.plattsburgh.edu/quenelgt/pubpdf/diamest.pdf ↩
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. http://www.mathe2.uni-bayreuth.de/markus/pdf/pub/FastGenRegGraphJGT.pdf ↩