Dipole graph

In graph theory, a dipole graph, dipole, bond graph, or linkage, is a multigraph consisting of two vertices connected with a number of parallel edges. A dipole graph containing n edges is called the size-n dipole graph, and is denoted by D_n. The size-n dipole graph is dual to the cycle graph C_n.

Dipole graph
Vertices	2
Edges	n
Diameter	1 (for n ≥ 1)
Chromatic number	2 (for n ≥ 1)
Chromatic index	n
Properties	connected (for n ≥ 1) planar
Table of graphs and parameters

The honeycomb as an abstract graph is the maximal abelian covering graph of the dipole graph D₃, while the diamond crystal as an abstract graph is the maximal abelian covering graph of D₄.

Similarly to the Platonic graphs, the dipole graphs form the skeletons of the hosohedra. Their duals, the cycle graphs, form the skeletons of the dihedra.

References

• Weisstein, Eric W. "Dipole Graph". MathWorld.
• Jonathan L. Gross and Jay Yellen, 2006. Graph Theory and Its Applications, 2nd Ed., p. 17. Chapman & Hall/CRC. ISBN 1-58488-505-X
• Sunada T., Topological Crystallography, With a View Towards Discrete Geometric Analysis, Springer, 2013, ISBN 978-4-431-54176-9 (Print) 978-4-431-54177-6 (Online)