The Desargues graph is the cubic symmetric graph on 20 vertices and 30 edges
illustrated above in several drawings. It is isomorphic to the generalized
Petersen graph 
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 
.
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. 2020).
The Desargues graph 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).
This graph is implemented in the Wolfram Language as GraphData["DesarguesGraph"].
."