Danzer Graph


The Danzer graph is the Levi graph of the Danzer configuration (Boben et al. 2015). It has 70 vertices and 140 edges and is quartic, bipartite, self-dual, and unit-distance. It is illustrated above in a unit-distance embedding (left) an LCF embedding of order 5 (center; Boben et al. 2015), and a bilaterally symmetric LCF order-1 embedding (right; E. Pegg Jr., pers. comm., Oct. 30, 2022).

The Danzer graph is isomorphic to the middle layer graph of order 3 and the bipartite Kneser graph H(7,3). It is the bipartite double graph of the odd graph O_4.

The Danzer graph is distance-regular (Brouwer and Koolen 1999) and distance-transitive.

The Danzer graph is implemented in the Wolfram Language as GraphData["DanzerGraph"].

See also

Bipartite Kneser Graph, Danzer Configuration, Levi Graph, Middle Layer Graph

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Boben, M.; Gévay, G. Pisanski, T. "Danzer's Configuration Revisited." Adv. Geom. 15, 393-408, 2015.Brouwer, A. and Koolen, J. "The Distance-Regular Graphs of Valency Four." J. Algebraic Combin. 10, 5-24, 1999.Gévay, G. "Pascal's Triangle of Configurations." In Discrete Geometry and Symmetry (Ed. M. D. E. Conder, A. Deza, and A. I. Weiss). Springer, pp. 181-199, 2018.Grünbaum, B. "Musing on an Example of Danzer's." Europ. J. Combin. 29, 1910-1918, 2018.Mütze, T. "Proof of the Middle Levels Conjecture." Proc. Lond. Math. Soc. 112, 677-713, 2016.

Cite this as:

Weisstein, Eric W. "Danzer Graph." From MathWorld--A Wolfram Web Resource.

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