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 $Km$ is strongly regular for any $m$.

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.