Langbahn Team – Weltmeisterschaft

Regular graph: Difference between revisions

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a graph is connected and regular if and only if the matrix ''J'', with <math>J_{ij}=1</math>, is in the [[adjacency algebra]] of the graph (meaning it is a linear combination of powers of ''A'').{{fact|date=February 2009}}
a graph is connected and regular if and only if the matrix ''J'', with <math>J_{ij}=1</math>, is in the [[adjacency algebra]] of the graph (meaning it is a linear combination of powers of ''A'').{{fact|date=February 2009}}


Let G be a k-regular graph with diameter d and eigenvalues of adjacency matrix <math>k=\lambda_0 >\lambda_1\geq \dots\geq\lambda_{n-1}</math>. If G is not bipartite
Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix <math>k=\lambda_0 >\lambda_1\geq \dots\geq\lambda_{n-1}</math>. If G is not bipartite


<math>d\leq \frac{\log{(n-1)}}{\log(k/\lambda)}+1</math>
<math>D\leq \frac{\log{(n-1)}}{\log(k/\lambda)}+1</math>


where <math> \lambda=\max_{i>0}\{\mid \lambda_i \mid \}</math>.{{fact|date=March 2009}}
where <math> \lambda=\max_{i>0}\{\mid \lambda_i \mid \}</math>.{{fact|date=March 2009}}

Revision as of 07:11, 21 April 2009

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 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 is strongly regular for any .

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

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.[1] Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to , so for such eigenvectors , we have .

When the graph is regular with degree k, the graph will be connected if and only if k has algebraic (and geometric) dimension one.[1]

There is also a criterion for regular and connected graphs : a graph is connected and regular if and only if the matrix J, with , is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).[citation needed]

Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix . If G is not bipartite

where .[citation needed]

Generation

Regular graphs may be generated by GenReg program. [2]

See also

References

  1. ^ a b Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.
  2. ^ M. Meringer, J. Graph Theory, 1999, 30, 137.