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.^{[1]} A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k.
Graph families defined by their automorphisms | ||||
---|---|---|---|---|
distance-transitive | → | distance-regular | ← | strongly regular |
↓ | ||||
symmetric (arc-transitive) | ← | t-transitive, t ≥ 2 | skew-symmetric | |
↓ | ||||
_{(if connected)} vertex- and edge-transitive |
→ | edge-transitive and regular | → | edge-transitive |
↓ | ↓ | ↓ | ||
vertex-transitive | → | regular | → | _{(if bipartite)} biregular |
↑ | ||||
Cayley graph | ← | zero-symmetric | asymmetric |
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.
The necessary and sufficient conditions for a -regular graph of order to exist are that and that 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 and degree here is . So . This is the minimum for a particular . Also note that if any regular graph has order then number of edges are so has to be even. In such case it is easy to construct regular graphs by considering appropriate parameters for circulant graphs.
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 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. The "only if" direction is a consequence of the Perron–Frobenius theorem.^{[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 , 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 . If G is not bipartite, then
Fast algorithms exist to generate, up to isomorphism, all regular graphs with a given degree and number of vertices.^{[5]}