Graph where each vertex has the same number of neighbors
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 graph or regular graph of degree k.
 ;templatestyles src="Template:TOC_left/styles.css" / ;
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 formula_1-regular graph of order formula_2 to exist are that formula_3 and that formula_4 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 formula_5 and degree here is formula_6. So formula_7. This is the minimum formula_2 for a particular formula_1. Also note that if any regular graph has order formula_2 then number of edges are formula_11 so formula_12 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 graph on 2"k" + 1 vertices has a Hamiltonian cycle.
Let "A" be the adjacency matrix of a graph. Then the graph is regular if and only if formula_13 is an eigenvector of "A". Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to formula_14, so for such eigenvectors formula_15, we have formula_16.
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 formula_17, 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 formula_18. If "G" is not bipartite, then
 formula_19
Generation.
Fast algorithms exist to generate, up to isomorphism, all regular graphs with a given degree and number of vertices.
References.
 ;templatestyles src="Reflist/styles.css" / ;