The Desargues graph is one of three cubic graphs on 20 nodes with smallest possible graph crossing
number of 6 (the others being two unnamed graphs denoted CNG 6B and CNG 6C by
Pegg and Exoo 2009), making it a smallest
cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).
The Desargues is an integral graph with graph spectrum.
It is cospectral with another nonisomorphic
graph (Haemers and Spence 1995, van Dam and Haemers 2003).
It is also a unit-distance graph (Gerbracht
2008) and is 3-unitransitive (Harary 1994, p. 175).
The Desargues graph is the first of four graphs depicted on the cover of Harary (1994).
Brouwer, A. E. "Desargues Graph." http://www.win.tue.nl/~aeb/drg/graphs/Desargues.html.Brouwer, A. E. and Haemers, W. H. "The Gewirtz Graph: An Exercise in the Theory
of Graph Spectra." European J. Combin.14, 397-407, 1993.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. "Desargues
graph ."
http://www.distanceregular.org/graphs/desargues.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.Haemers,
W. H. and Spence, E. "Graphs Cospectral with Distance-Regular Graphs."
Linear Multilin. Alg.39, 91-107, 1995.Harary, F. Graph
Theory. Reading, MA: Addison-Wesley, 1994.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. "F020B." http://www.csse.uwa.edu.au/~gordon/foster/F020B.html.Royle,
G. "Cubic Symmetric Graphs (The Foster Census): Distance-Regular Graphs."
http://school.maths.uwa.edu.au/~gordon/remote/foster/#drgs.van
Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some
Distance-Regular Graphs." J. Algebraic Combin.15, 189-202, 2003.Wolfram,
S. A
New Kind of Science. Champaign, IL: Wolfram Media, p. 1032,
2002.
and to the bipartite
Kneser graph 
.
It is the Levi graph of the Desargues
configuration. It can be represented in LCF notation
as ![[5,-5,9,-9]^5](/images/equations/DesarguesGraph/Inline3.svg)
(Frucht 1976). It can also be constructed as the graph
expansion of 
with steps 1 and 3, where 
is a path graph. It is distance-transitive and distance-regular
graph and has intersection array 
.
.
It is cospectral with another nonisomorphic
graph (Haemers and Spence 1995, van Dam and Haemers 2003).
."