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Pappus Graph

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PappusGraphEmbeddings

The Pappus graph is a cubic symmetric distance-regular graph on 18 vertices, illustrated above in three embeddings. It is Hamiltonian and can be represented in LCF notation as [5,7,-7,7,-7,-5]^3 (Frucht 1976). It is the Levi graph of the 9_3 configuration appearing in Pappus's hexagon theorem, namely the Pappus configuration. It is also Bouwer graph B(3,2,3) and honeycomb toroidal graph HTG(3,6,3).

The Pappus graph is one of two cubic graphs on 18 nodes with smallest possible graph crossing number of 5 (the other being an unnamed graph denoted CNG 5B by Pegg and Exoo 2009), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).

PappusGraphUnitDistance

It is also a unit-distance graph, as illustrated in the above embedding (Gerbracht 2008; E. Gerbracht, pers. comm., Jan. 2, 2010).

PappusGraphMatrices

The plots above show the adjacency, incidence, and graph distance matrices for the Pappus graph.

The graph spectrum of the Pappus graph is (-3)^1(-sqrt(3))^60^4(sqrt(3))^63^1.

See also

Cubic Symmetric Graph, Distance-Regular Graph, Honeycomb Toroidal Graph, Pappus Configuration, Pappus's Hexagon Theorem, Smallest Cubic Crossing Number Graph

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References

Berman, L. W.; Gévay, G.; and Pisanski, T. "Polycyclic Geometric Realizations of the Gray Configuration." 20 Feb 2025. https://arxiv.org/abs/2502.14484.Brouwer, A. E. "Pappus Graph." http://www.win.tue.nl/~aeb/drg/graphs/Pappus.html.Clancy, K.; Haythorpe, M.; Newcombe, A.; and Pegg, E. Jr. "There Are No Cubic Graphs on 26 Vertices with Crossing Number 10 or 11." Preprint. 2019.Coxeter, H. S. M. "Self-Dual Configurations and Regular Graphs." Bull. Amer. Math. Soc. 56, 413-455, 1950.DistanceRegular.org. "Pappus Graph. = Incidence Graph of AG(2,3) Minus a Parallel Class" https://www.math.mun.ca/distanceregular/graphs//pappus.html.Frucht, R. "A Canonical Representation of Trivalent Hamiltonian Graphs." J. Graph Th. 1, 45-60, 1976.Gerbracht, E. H.-A. "On the Unit Distance Embeddability of Connected Cubic Symmetric Graphs." Kolloquium über Kombinatorik. Magdeburg, Germany. Nov. 15, 2008.Kagno, I. N. "Desargues' and Pappus' Graphs and Their Groups." Amer. J. Math. 69, 859-863, 1947.Pegg, E. Jr. and Exoo, G. "Crossing Number Graphs." Mathematica J. 11, 161-170, 2009. https://www.mathematica-journal.com/data/uploads/2009/11/CrossingNumberGraphs.pdf.Royle, G. "F018A." http://www.csse.uwa.edu.au/~gordon/foster/F018A.html.Royle, G. "Cubic Symmetric Graphs (The Foster Census): Distance-Regular Graphs." http://school.maths.uwa.edu.au/~gordon/remote/foster/#drgs.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1032, 2002.

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Pappus Graph

Cite this as:

Weisstein, Eric W. "Pappus Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PappusGraph.html

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