Regular graph

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In graph theory, a regular graph is a graph without loops and multiple edges 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.

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.

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

0-regular graph

1-regular graph

2-regular graph

3-regular graph


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]

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