Berman-Gévay-Pisanski Graph

Author

Eric W. Weisstein
July 31, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/Berman-Gevay-PisanskiGraph.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Berman-Gevay-PisanskiGraph.html.

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

Sources

Levi graph of the Berman-Gevay-Pisanski configuration. Source: L. W. Berman, G. Gevay, and T. Pisanski, On a New (21_4) Polycyclic Configuration, Electronic Journal of Combinatorics 31 (2024), P4.54, https://doi.org/10.37236/12405; https://arxiv.org/abs/2309.12992.

Incidence data

incidence={{"M0","M1","G1","P0"},{"M1","M2","G2","P1"},{"M2","M0","G0","P2"},{"M2","B1","C0","P0"},{"M0","B2","C1","P1"},{"M1","B0","C2","P2"},{"B0","C1","R1","P0"},{"B1","C2","R2","P1"},{"B2","C0","R0","P2"},{"Y2","R0","R1","P1"},{"Y0","R1","R2","P2"},{"Y1","R2","R0","P0"},{"C0","C2","Y1","G0"},{"C1","C0","Y2","G1"},{"C2","C1","Y0","G2"},{"B0","B2","G1","Y0"},{"B1","B0","G2","Y1"},{"B2","B1","G0","Y2"},{"R0","Y0","G0","M1"},{"R1","Y1","G1","M2"},{"R2","Y2","G2","M0"}}

Levi graph

edges={"Point \"M0\""\[UndirectedEdge]"Line 1","Point \"M1\""\[UndirectedEdge]"Line 1","Point \"G1\""\[UndirectedEdge]"Line 1","Point \"P0\""\[UndirectedEdge]"Line 1","Point \"M1\""\[UndirectedEdge]"Line 2","Point \"M2\""\[UndirectedEdge]"Line 2","Point \"G2\""\[UndirectedEdge]"Line 2","Point \"P1\""\[UndirectedEdge]"Line 2","Point \"M2\""\[UndirectedEdge]"Line 3","Point \"M0\""\[UndirectedEdge]"Line 3","Point \"G0\""\[UndirectedEdge]"Line 3","Point \"P2\""\[UndirectedEdge]"Line 3","Point \"M2\""\[UndirectedEdge]"Line 4","Point \"B1\""\[UndirectedEdge]"Line 4","Point \"C0\""\[UndirectedEdge]"Line 4","Point \"P0\""\[UndirectedEdge]"Line 4","Point \"M0\""\[UndirectedEdge]"Line 5","Point \"B2\""\[UndirectedEdge]"Line 5","Point \"C1\""\[UndirectedEdge]"Line 5","Point \"P1\""\[UndirectedEdge]"Line 5","Point \"M1\""\[UndirectedEdge]"Line 6","Point \"B0\""\[UndirectedEdge]"Line 6","Point \"C2\""\[UndirectedEdge]"Line 6","Point \"P2\""\[UndirectedEdge]"Line 6","Point \"B0\""\[UndirectedEdge]"Line 7","Point \"C1\""\[UndirectedEdge]"Line 7","Point \"R1\""\[UndirectedEdge]"Line 7","Point \"P0\""\[UndirectedEdge]"Line 7","Point \"B1\""\[UndirectedEdge]"Line 8","Point \"C2\""\[UndirectedEdge]"Line 8","Point \"R2\""\[UndirectedEdge]"Line 8","Point \"P1\""\[UndirectedEdge]"Line 8","Point \"B2\""\[UndirectedEdge]"Line 9","Point \"C0\""\[UndirectedEdge]"Line 9","Point \"R0\""\[UndirectedEdge]"Line 9","Point \"P2\""\[UndirectedEdge]"Line 9","Point \"Y2\""\[UndirectedEdge]"Line 10","Point \"R0\""\[UndirectedEdge]"Line 10","Point \"R1\""\[UndirectedEdge]"Line 10","Point \"P1\""\[UndirectedEdge]"Line 10","Point \"Y0\""\[UndirectedEdge]"Line 11","Point \"R1\""\[UndirectedEdge]"Line 11","Point \"R2\""\[UndirectedEdge]"Line 11","Point \"P2\""\[UndirectedEdge]"Line 11","Point \"Y1\""\[UndirectedEdge]"Line 12","Point \"R2\""\[UndirectedEdge]"Line 12","Point \"R0\""\[UndirectedEdge]"Line 12","Point \"P0\""\[UndirectedEdge]"Line 12","Point \"C0\""\[UndirectedEdge]"Line 13","Point \"C2\""\[UndirectedEdge]"Line 13","Point \"Y1\""\[UndirectedEdge]"Line 13","Point \"G0\""\[UndirectedEdge]"Line 13","Point \"C1\""\[UndirectedEdge]"Line 14","Point \"C0\""\[UndirectedEdge]"Line 14","Point \"Y2\""\[UndirectedEdge]"Line 14","Point \"G1\""\[UndirectedEdge]"Line 14","Point \"C2\""\[UndirectedEdge]"Line 15","Point \"C1\""\[UndirectedEdge]"Line 15","Point \"Y0\""\[UndirectedEdge]"Line 15","Point \"G2\""\[UndirectedEdge]"Line 15","Point \"B0\""\[UndirectedEdge]"Line 16","Point \"B2\""\[UndirectedEdge]"Line 16","Point \"G1\""\[UndirectedEdge]"Line 16","Point \"Y0\""\[UndirectedEdge]"Line 16","Point \"B1\""\[UndirectedEdge]"Line 17","Point \"B0\""\[UndirectedEdge]"Line 17","Point \"G2\""\[UndirectedEdge]"Line 17","Point \"Y1\""\[UndirectedEdge]"Line 17","Point \"B2\""\[UndirectedEdge]"Line 18","Point \"B1\""\[UndirectedEdge]"Line 18","Point \"G0\""\[UndirectedEdge]"Line 18","Point \"Y2\""\[UndirectedEdge]"Line 18","Point \"R0\""\[UndirectedEdge]"Line 19","Point \"Y0\""\[UndirectedEdge]"Line 19","Point \"G0\""\[UndirectedEdge]"Line 19","Point \"M1\""\[UndirectedEdge]"Line 19","Point \"R1\""\[UndirectedEdge]"Line 20","Point \"Y1\""\[UndirectedEdge]"Line 20","Point \"G1\""\[UndirectedEdge]"Line 20","Point \"M2\""\[UndirectedEdge]"Line 20","Point \"R2\""\[UndirectedEdge]"Line 21","Point \"Y2\""\[UndirectedEdge]"Line 21","Point \"G2\""\[UndirectedEdge]"Line 21","Point \"M0\""\[UndirectedEdge]"Line 21"}

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]]}