Regular graph
Template:Short description Template:Refimprove 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.<ref> Template:Cite book</ref> A regular graph with vertices of degree Template:Mvar is called a Template:Nowrap graph or regular graph of degree Template:Mvar.
Special cases
Regular graphs of degree at most 2 are easy to classify: a Template:Nowrap graph consists of disconnected vertices, a Template:Nowrap graph consists of disconnected edges, and a Template:Nowrap graph consists of a disjoint union of cycles and infinite chains.
In analogy with the terminology for polynomials of low degrees, a Template:Nowrap or Template:Nowrap graph often is called a cubic graph or a quartic graph, respectively. Similarly, it is possible to denote k-regular graphs with <math>k=5,6,7,8,\ldots</math> 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 Template:Mvar of neighbors in common, and every non-adjacent pair of vertices has the same number Template:Mvar 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 Template:Mvar is strongly regular for any Template:Mvar.
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0-regular graph -
1-regular graph -
2-regular graph -
3-regular graph
Properties
By the degree sum formula, a Template:Mvar-regular graph with Template:Mvar vertices has <math>\frac{nk}2</math> edges. In particular, at least one of the order Template:Mvar and the degree Template:Mvar must be an even number.
A theorem by Nash-Williams says that every Template:Nowrap graph on Template:Math vertices has a Hamiltonian cycle.
Let A be the adjacency matrix of a graph. Then the graph is regular if and only if <math>\textbf{j}=(1, \dots ,1)</math> is an eigenvector of A.<ref name="Cvetkovic">Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.</ref> Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to <math>\textbf{j}</math>, so for such eigenvectors <math>v=(v_1,\dots,v_n)</math>, we have <math>\sum_{i=1}^n v_i = 0</math>.
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.<ref name="Cvetkovic"/>
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 <math>J_{ij}=1</math>, is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).<ref>Template:Citation.</ref>
Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix <math>k=\lambda_0 >\lambda_1\geq \cdots\geq\lambda_{n-1}</math>. If G is not bipartite, then
- <math>D\leq \frac{\log{(n-1)}}{\log(\lambda_0/\lambda_1)}+1. </math><ref>Template:Cite journal[1]</ref>
Existence
There exists a <math>k</math>-regular graph of order <math>n</math> if and only if the natural numbers Template:Mvar and Template:Mvar satisfy the inequality <math> n \geq k+1 </math> and that <math> nk </math> is even.
Proof: If a graph with Template:Mvar vertices is Template:Mvar-regular, then the degree Template:Mvar of any vertex v cannot exceed the number <math>n-1</math> of vertices different from v, and indeed at least one of Template:Mvar and Template:Mvar must be even, whence so is their product.
Conversely, if Template:Mvar and Template:Mvar are two natural numbers satisfying both the inequality and the parity condition, then indeed there is a Template:Mvar-regular circulant graph <math>C_n^{s_1,\ldots,s_r}</math> of order Template:Mvar (where the <math>s_i</math> denote the minimal `jumps' such that vertices with indices differing by an <math>s_i</math> are adjacent). If in addition Template:Mvar is even, then <math>k = 2r</math>, and a possible choice is <math>(s_1,\ldots,s_r) = (1,2,\ldots,r)</math>. Else Template:Mvar is odd, whence Template:Mvar must be even, say with <math>n = 2m</math>, and then <math>k = 2r-1</math> and the `jumps' may be chosen as <math>(s_1,\ldots,s_r) = (1,2,\ldots,r-1,m)</math>.
If <math>n=k+1</math>, 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.<ref>Template:Cite journal</ref>
See also
References
External links
- Template:MathWorld
- Template:MathWorld
- GenReg software and data by Markus Meringer.
- Template:Citation