Regular graph

Properties

By the degree sum formula, a k-regular graph with n vertices has ${\displaystyle \frac{nk}{2}}$ 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 ${\displaystyle \mathbf{\text{j}}=(1,\dots ,1)}$ is an eigenvector of A. Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to ${\displaystyle \mathbf{\text{j}}}$, so for such eigenvectors ${\displaystyle v=(v_1,\dots ,v_n)}$, we have ${\displaystyle \sum _{i=1}^n{v}_i=0}$.

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 ${\displaystyle J_{ij}=1}$, 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 ${\displaystyle k={\lambda }_0>{\lambda }_1\ge \cdots \ge {\lambda }_{n-1}}$. If G is not bipartite, then

${\displaystyle D\le \frac{\log (n-1)}{\log ({\lambda }_0/{\lambda }_1)}+1.}$

Existence

There exists a ${\displaystyle k}$-regular graph of order ${\displaystyle n}$ if and only if the natural numbers n and k satisfy the inequality ${\displaystyle n\ge k+1}$ and that ${\displaystyle nk}$ is even.

Proof: If a graph with n vertices is k-regular, then the degree k of any vertex v cannot exceed the number ${\displaystyle n-1}$ 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 ${\displaystyle C_n^{s_1,\dots ,s_r}}$ of order n (where the ${\displaystyle s_i}$ denote the minimal `jumps' such that vertices with indices differing by an ${\displaystyle s_i}$ are adjacent). If in addition k is even, then ${\displaystyle k=2r}$, and a possible choice is ${\displaystyle (s_1,\dots ,s_r)=(1,2,\dots ,r)}$. Else k is odd, whence n must be even, say with ${\displaystyle n=2m}$, and then ${\displaystyle k=2r-1}$ and the `jumps' may be chosen as ${\displaystyle (s_1,\dots ,s_r)=(1,2,\dots ,r-1,m)}$.

If ${\displaystyle n=k+1}$, 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.