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 kâ€‘regular graph or regular graph of degree k.
Regular graphs of degree at most 2 are easy to classify: A 0regular graph consists of disconnected vertices, a 1regular graph consists of disconnected edges, and a 2regular graph consists of disconnected cycles.
A 3regular 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 nonadjacent 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 NashWilliams says that every kâ€‘regular graph on 2k + 1 vertices has a Hamiltonian cycle.
0regular graph
1regular graph
2regular graph
3regular graph
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Algebraic properties
Let A be the adjacency matrix of a graph. Then the graph is regular if and only if is an eigenvector of A.^{[2]} Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to , so for such eigenvectors , we have .
A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one.^{[2]}
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