[1] It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level of symmetry, is the only known non-planar cubic partial cube, and has been applied in chemical databases.

One can interpret this product representation of the symmetry group in terms of the constructions of the Desargues graph: the symmetric group on five points is the symmetry group of the Desargues configuration, and the order-2 subgroup swaps the roles of the vertices that represent points of the Desargues configuration and the vertices that represent lines.

The generalized Petersen graph G(n, k) is vertex-transitive if and only if n = 10 and k = 2 or if k2 ≡ ±1 (mod n) and is edge-transitive only in the following seven cases: (n, k) = (4, 1), (5, 2), (8, 3), (10, 2), (10, 3), (12, 5), (24, 5).

In this application, the thirty edges of the graph correspond to pseudorotations of the ligands.

The Desargues graph can be embedded as a self-Petrie dual regular map in the non-orientable manifold of genus 6, with decagonal faces.