# 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. 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 K$_m$ is strongly regular for any m.
* 0-regular graph
* 1-regular graph
* 2-regular graph
* 3-regular graph

## Existence

The necessary and sufficient conditions for a $k$-regular graph of order $n$ to exist are that $n\geq k+1$ and that $nk$ 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 ${\binom {n}{2}}={\dfrac {n(n-1)}{2}}$ and degree here is $n-1$. So $k=n-1,n=k+1$. This is the minimum $n$ for a particular $k$. Also note that if any regular graph has order $n$ then number of edges are ${\dfrac {nk}{2}}$ so $nk$ 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 ${\textbf {j}}=(1,\dots ,1)$ is an eigenvector of A. Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to ${\textbf {j}}$, so for such eigenvectors $v=(v_{1},\dots ,v_{n})$, we have $\sum _{i=1}^{n}v_{i}=0$.
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 $J_{ij}=1$, 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 $k=\lambda _{0}>\lambda _{1}\geq \cdots \geq \lambda _{n-1}$. If G is not bipartite, then
    $D\leq {\frac {\log {(n-1)}}{\log(\lambda _{0}/\lambda _{1})}}+1.$

## Generation

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