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](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FPappusGraph%2FInline1.svg){kind=small}
(Frucht 1976). It is the Levi
graph of the 
configuration appearing in Pappus's
hexagon theorem, namely the Pappus configuration.
It is also Bouwer graph 
and honeycomb
toroidal graph 
.
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).
It is also a unit-distance graph, as illustrated in the above embedding (Gerbracht 2008; E. Gerbracht, pers. comm., Jan. 2, 2010).
The plots above show the adjacency, incidence, and graph distance matrices for the Pappus graph.
The graph spectrum of the Pappus graph is 
.
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 
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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Pappus Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PappusGraph.html