Regular graph

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.

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 ${\displaystyle k=5,6,7,8,\ldots }${\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

By the degree sum formula, a k-regular graph with n vertices has ${\displaystyle {\frac {nk}{2}}}${\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 ${\displaystyle {\textbf {j}}=(1,\dots ,1)}${\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 ${\displaystyle {\textbf {j}}}${\displaystyle {\textbf {j}}}, so for such eigenvectors ${\displaystyle v=(v_{1},\dots ,v_{n})}${\displaystyle v=(v_{1},\dots ,v_{n})}, we have ${\displaystyle \sum _{i=1}^{n}v_{i}=0}${\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.^[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 ${\displaystyle J_{ij}=1}${\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 ${\displaystyle k=\lambda _{0}>\lambda _{1}\geq \cdots \geq \lambda _{n-1}}${\displaystyle k=\lambda _{0}>\lambda _{1}\geq \cdots \geq \lambda _{n-1}}. If G is not bipartite, then

${\displaystyle D\leq {\frac {\log {(n-1)}}{\log(\lambda _{0}/\lambda _{1})}}+1.}${\displaystyle D\leq {\frac {\log {(n-1)}}{\log(\lambda _{0}/\lambda _{1})}}+1.}^[4]

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

If ${\displaystyle n=k+1}${\displaystyle n=k+1}, then this circulant graph is complete.

Fast algorithms exist to generate, up to isomorphism, all regular graphs with a given degree and number of vertices.^[5]

• Random regular graph
• Strongly regular graph
• Moore graph
• Cage graph
• Highly irregular graph

• 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

• Graph families
• Regular graphs

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