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 , … {\displaystyle k=5,6,7,8,\ldots } as quintic, sextic, septic, octic, etcetera.

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.

  • 0-regular graph
  • 1-regular graph
  • 2-regular graph
  • 3-regular graph

Properties

By the degree sum formula, a k-regular graph with n vertices has n k 2 {\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 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. 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.

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).

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.}

Existence

There exists a k {\displaystyle k}-regular graph of order n {\displaystyle n} if and only if the natural numbers n and k satisfy the inequality n ≥ k + 1 {\displaystyle n\geq k+1} and that n k {\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 n − 1 {\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 C n s 1 , … , s r {\displaystyle C_{n}^{s_{1},\ldots ,s_{r}}} of order n (where the s i {\displaystyle s_{i}} denote the minimal `jumps' such that vertices with indices differing by an s i {\displaystyle s_{i}} are adjacent). If in addition k is even, then k = 2 r {\displaystyle k=2r}, and a possible choice is ( s 1 , … , s r ) = ( 1 , 2 , … , r ) {\displaystyle (s_{1},\ldots ,s_{r})=(1,2,\ldots ,r)}. Else k is odd, whence n must be even, say with n = 2 m {\displaystyle n=2m}, and then k = 2 r − 1 {\displaystyle k=2r-1} and the `jumps' may be chosen as ( s 1 , … , s r ) = ( 1 , 2 , … , r − 1 , m ) {\displaystyle (s_{1},\ldots ,s_{r})=(1,2,\ldots ,r-1,m)}.

If n = k + 1 {\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.

See also

External links