Dipole graph

Dipole graph
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

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 hexagonal tiling as an abstract graph is the maximal abelian covering graph of the dipole graph $D 3$ , while the diamond crystal as an abstract graph is the maximal abelian covering graph of $D 4$ .

The dipole graph can be embedded on a sphere, subdividing the sphere into a hosohedron, consisting only of two-sided digons. This subdivision is a regular map, a generalization of the Platonic solids. The dual graph of the size- $n$ dipole graph, embedded in this way, is the cycle graph $C n$ , embedded as a dihedron.

References

↑ Gross, Jonathan L.; Yellen, Jay; Anderson, Mark (2019), Graph Theory and its Applications, Textbooks in Mathematics (3rd ed.), Boca Raton, Florida: CRC Press, pp. 17–18, ISBN 978-1-4822-4948-4, MR 3887696
↑ Sunada, Toshikazu (2013), Topological Crystallography, With a View Towards Discrete Geometric Analysis, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 6, Tokyo: Springer, p. 2, 7, doi:10.1007/978-4-431-54177-6, ISBN 978-4-431-54176-9, MR 3014418
↑ Nedela, Roman; Škoviera, Martin (1996), "Regular embeddings of canonical double coverings of graphs", Journal of Combinatorial Theory, Series B, 67 (2): 249–277, doi:10.1006/jctb.1996.0044, MR 1399678
↑ Weisstein, Eric W., "Dipole Graph", MathWorld

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[1] Gross, Jonathan L.; Yellen, Jay; Anderson, Mark (2019), Graph Theory and its Applications, Textbooks in Mathematics (3rd ed.), Boca Raton, Florida: CRC Press, pp. 17–18, ISBN 978-1-4822-4948-4, MR 3887696

[2] Sunada, Toshikazu (2013), Topological Crystallography, With a View Towards Discrete Geometric Analysis, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 6, Tokyo: Springer, p. 2, 7, doi:10.1007/978-4-431-54177-6, ISBN 978-4-431-54176-9, MR 3014418

[3] Nedela, Roman; Škoviera, Martin (1996), "Regular embeddings of canonical double coverings of graphs", Journal of Combinatorial Theory, Series B, 67 (2): 249–277, doi:10.1006/jctb.1996.0044, MR 1399678

[4] Weisstein, Eric W., "Dipole Graph", MathWorld