Bokowski-Pilaud Graph

Author

Eric W. Weisstein
July 31, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/Bokowski-PilaudGraph.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Bokowski-PilaudGraph.html.

©2026 Wolfram Research, Inc. except for portions noted otherwise

Sources

Levi graph of the Bokowski-Pilaud configuration. Source: J. Bokowski and V. Pilaud, Enumerating Topological (n_k)-Configurations, Computational Geometry 47 (2014), 175-186, https://doi.org/10.1016/j.comgeo.2012.10.002; https://arxiv.org/abs/1210.0306.

Incidence data

incidence={{13,14,15,16},{9,12,16,17},{4,8,7,16},{3,6,5,16},{4,15,11,17},{4,12,10,18},{3,11,9,18},{3,8,10,15},{14,9,7,2},{14,6,8,18},{13,5,7,18},{13,2,6,17},{14,11,12,1},{13,9,10,1},{8,5,1,17},{4,3,2,1},{2,5,12,15},{6,7,10,11}}

Levi graph

edges={"Point 13"\[UndirectedEdge]"Line 1","Point 14"\[UndirectedEdge]"Line 1","Point 15"\[UndirectedEdge]"Line 1","Point 16"\[UndirectedEdge]"Line 1","Point 9"\[UndirectedEdge]"Line 2","Point 12"\[UndirectedEdge]"Line 2","Point 16"\[UndirectedEdge]"Line 2","Point 17"\[UndirectedEdge]"Line 2","Point 4"\[UndirectedEdge]"Line 3","Point 8"\[UndirectedEdge]"Line 3","Point 7"\[UndirectedEdge]"Line 3","Point 16"\[UndirectedEdge]"Line 3","Point 3"\[UndirectedEdge]"Line 4","Point 6"\[UndirectedEdge]"Line 4","Point 5"\[UndirectedEdge]"Line 4","Point 16"\[UndirectedEdge]"Line 4","Point 4"\[UndirectedEdge]"Line 5","Point 15"\[UndirectedEdge]"Line 5","Point 11"\[UndirectedEdge]"Line 5","Point 17"\[UndirectedEdge]"Line 5","Point 4"\[UndirectedEdge]"Line 6","Point 12"\[UndirectedEdge]"Line 6","Point 10"\[UndirectedEdge]"Line 6","Point 18"\[UndirectedEdge]"Line 6","Point 3"\[UndirectedEdge]"Line 7","Point 11"\[UndirectedEdge]"Line 7","Point 9"\[UndirectedEdge]"Line 7","Point 18"\[UndirectedEdge]"Line 7","Point 3"\[UndirectedEdge]"Line 8","Point 8"\[UndirectedEdge]"Line 8","Point 10"\[UndirectedEdge]"Line 8","Point 15"\[UndirectedEdge]"Line 8","Point 14"\[UndirectedEdge]"Line 9","Point 9"\[UndirectedEdge]"Line 9","Point 7"\[UndirectedEdge]"Line 9","Point 2"\[UndirectedEdge]"Line 9","Point 14"\[UndirectedEdge]"Line 10","Point 6"\[UndirectedEdge]"Line 10","Point 8"\[UndirectedEdge]"Line 10","Point 18"\[UndirectedEdge]"Line 10","Point 13"\[UndirectedEdge]"Line 11","Point 5"\[UndirectedEdge]"Line 11","Point 7"\[UndirectedEdge]"Line 11","Point 18"\[UndirectedEdge]"Line 11","Point 13"\[UndirectedEdge]"Line 12","Point 2"\[UndirectedEdge]"Line 12","Point 6"\[UndirectedEdge]"Line 12","Point 17"\[UndirectedEdge]"Line 12","Point 14"\[UndirectedEdge]"Line 13","Point 11"\[UndirectedEdge]"Line 13","Point 12"\[UndirectedEdge]"Line 13","Point 1"\[UndirectedEdge]"Line 13","Point 13"\[UndirectedEdge]"Line 14","Point 9"\[UndirectedEdge]"Line 14","Point 10"\[UndirectedEdge]"Line 14","Point 1"\[UndirectedEdge]"Line 14","Point 8"\[UndirectedEdge]"Line 15","Point 5"\[UndirectedEdge]"Line 15","Point 1"\[UndirectedEdge]"Line 15","Point 17"\[UndirectedEdge]"Line 15","Point 4"\[UndirectedEdge]"Line 16","Point 3"\[UndirectedEdge]"Line 16","Point 2"\[UndirectedEdge]"Line 16","Point 1"\[UndirectedEdge]"Line 16","Point 2"\[UndirectedEdge]"Line 17","Point 5"\[UndirectedEdge]"Line 17","Point 12"\[UndirectedEdge]"Line 17","Point 15"\[UndirectedEdge]"Line 17","Point 6"\[UndirectedEdge]"Line 18","Point 7"\[UndirectedEdge]"Line 18","Point 10"\[UndirectedEdge]"Line 18","Point 11"\[UndirectedEdge]"Line 18"}

leviGraph=Graph[edges,GraphLayout->"BipartiteEmbedding"]

Checks

The result gives the vertex count, edge count, degree counts, bipartiteness, connectedness, and automorphism-group order.

{VertexCount[leviGraph],EdgeCount[leviGraph],Counts[VertexDegree[leviGraph]],BipartiteGraphQ[leviGraph],ConnectedGraphQ[leviGraph],GroupOrder[GraphAutomorphismGroup[leviGraph]]}