Regular Graphs

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. A regular graph with vertices of degree is called a graph or regular graph of degree . Also, from the handshaking lemma, a regular graph contains an even number of vertices with odd degree.

Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of cycles and infinite chains.

A 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 of neighbors in common, and every non-adjacent pair of vertices has the same number 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 is strongly regular for any .

A theorem by Nash-Williams says that every graph on vertices has a Hamiltonian cycle.

Image:0-regular_graph.svg, 0-regular graph
Image:1-regular_graph.svg, 1-regular graph
Image:2-regular_graph.svg, 2-regular graph
Image:3-regular_graph.svg, 3-regular graph

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.

Proof: As we know 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 $\binom = \dfrac$ and degree here is $n-1$. So $k=n-1,n=k+1$. This is the minimum $n$ for a particular $k$. Also note that if any regular graph has order $n$ then number of edges are $\dfrac$ so $nk$ has to be 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 $\textbf=(1, \dots ,1)$ is an eigenvector of ''A''.Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998. Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to $\textbf$, so for such eigenvectors $v=(v_1,\dots,v_n)$, we have $\sum_^n v_i = 0$.

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 $J_=1$, 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 $k=\lambda_0 >\lambda_1\geq \cdots\geq\lambda_$. If ''G'' is not bipartite, then

: $D\leq \frac+1.$

Generation

Fast algorithms exist to enumerate, up to isomorphism, all regular graphs with a given degree and number of vertices.

See also

* Random regular graph
* Strongly regular graph
* Moore graph
* Cage graph
* Highly irregular graph

References

External links

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GenReg
software and data by Markus Meringer.
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Graph families
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