# Regular graph

 Graph families defined by their automorphisms distance-transitive $\boldsymbol{\rightarrow}$ distance-regular $\boldsymbol{\leftarrow}$ strongly regular $\boldsymbol{\downarrow}$ symmetric (arc-transitive) $\boldsymbol{\leftarrow}$ t-transitive, t ≥ 2 $\boldsymbol{\downarrow}$ (if connected) vertex- and edge-transitive $\boldsymbol{\rightarrow}$ edge-transitive and regular $\boldsymbol{\rightarrow}$ edge-transitive $\boldsymbol{\downarrow}$ $\boldsymbol{\downarrow}$ $\boldsymbol{\downarrow}$ vertex-transitive $\boldsymbol{\rightarrow}$ regular $\boldsymbol{\rightarrow}$ (if bipartite) biregular $\boldsymbol{\uparrow}$ Cayley graph skew-symmetric asymmetric

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 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 disconnected 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 $m$.

A theorem by Nash-Williams says that every k‑regular graph on 2k + 1 vertices has a Hamiltonian cycle.

## Existence

It is well known that 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. In such case it is easy to construct regular graphs by considering appropriate parameters for circulant graphs.

## Algebraic properties

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.[2] 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.[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 $J_{ij}=1$, 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 $k=\lambda_0 >\lambda_1\geq \dots\geq\lambda_{n-1}$. If G is not bipartite

$D\leq \frac{\log{(n-1)}}{\log(k/\lambda)}+1$[4]

where $\lambda=\max_{i>0}\{\mid \lambda_i \mid \}$.[4]

## Generation

Regular graphs may be generated by the GenReg program.[5]