Desargues graph
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| Desargues graph | |
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| Named after | Gérard Desargues |
|---|---|
| Vertices | 20 |
| Edges | 30 |
| Girth | 6 |
| Chromatic number | 2 |
| Properties | Cubic Distance-regular |
In the mathematical field of graph theory, the Desargues graph is a 3-regular graph with 20 vertices and 30 edges, formed as the Levi graph of the Desargues configuration. It is a distance-regular graph, one of only 14 such cubic graphs according to Cubic symmetric graphs (The Foster Census). The Desargues graph can also be formed as a double cover of the Petersen graph, as the generalized Petersen graph G(10,3), or as the bipartite Kneser graph H5,2. It has crossing number 6, and is the smallest cubic graph with that crossing number (sequence A110507 in OEIS). It is the only known nonplanar cubic partial cube (Klavžar & Lipovec 2003).
In chemistry, the Desargues graph is known as the Desargues–Levi graph; it is used to organize systems of stereoisomers of 5-ligand compounds. In this application, the thirty edges of the graph correspond to pseudorotations of the ligands (Balaban, Fǎrcaşiu & Bǎnicǎ 1966; Mislow 1970).
The name "Desargues graph" has also been used to refer to the complement of the Petersen graph (Kagno 1947)).
[edit] References
- Balaban, A. T.; Fǎrcaşiu, D. & Bǎnicǎ, R. (1966), “Graphs of multiple 1, 2-shifts in carbonium ions and related systems”, Rev. Roum. Chim. 11: 1205.
- Kagno, I. N. (1947), “Desargues' and Pappus' graphs and their groups”, American Journal of Mathematics 69 (4): 859–863, DOI 10.2307/2371806.
- Klavžar, Sandi & Lipovec, Alenka (2003), “Partial cubes as subdivision graphs and as generalized Petersen graphs”, Discrete Mathematics 263: 157–165, doi:10.1016/S0012-365X(02)00575-7, <http://www.ijp.si/ftp/pub/preprints/ps/2001/pp737.ps>.
- Mislow, Kurt (1970), “Role of pseudorotation in the stereochemistry of nucleophilic displacement reactions”, Acc. Chem. Res. 3 (10): 321–331, DOI 10.1021/ar50034a001.
