Editing Incidence structure (section)

=== Incidence graph (Levi graph) ===
[[Image:Fano plane-Levi graph.svg|thumb|[[Heawood graph]] with labeling]]
Each incidence structure {{mvar|C}} corresponds to a [[bipartite graph]] called the [[Levi graph]] or incidence graph of the structure. As any bipartite graph is two-colorable, the Levi graph can be given a black and white [[graph coloring|vertex coloring]], where black vertices correspond to points and white vertices correspond to lines of {{mvar|C}}. The edges of this graph correspond to the flags (incident point/line pairs) of the incidence structure. The original Levi graph was the incidence graph of the [[generalized quadrangle]] of order two (example 3 above),<ref>{{citation  | last = Levi | first = F. W. | author-link = Friedrich Wilhelm Levi  | location = Calcutta  | mr = 0006834  | publisher = University of Calcutta  | title = Finite Geometrical Systems  | year = 1942}}</ref> but the term has been extended by [[H.S.M. Coxeter]]<ref>{{citation|first=H.S.M.|last=Coxeter|author-link=H.S.M. Coxeter|title=Self-dual configurations and regular graphs|journal=Bulletin of the American Mathematical Society|volume=56|year=1950|issue=5 |pages=413–455|doi=10.1090/s0002-9904-1950-09407-5|doi-access=free}}</ref> to refer to an incidence graph of any incidence structure.<ref name=Pisanski158>{{harvnb|Pisanski|Servatius|2013|page=158}}</ref>
[[File:Möbius–Kantor unit distance.svg|left|thumb|Levi graph of the Möbius–Kantor configuration (#4)]]

====Levi graph examples ====

The Levi graph of the [[Fano plane]] is the [[Heawood graph]]. Since the Heawood graph is [[Connected graph|connected]] and [[vertex-transitive]], there exists an [[automorphism]] (such as the one defined by a reflection about the vertical axis in the figure of the Heawood graph) interchanging black and white vertices. This, in turn, implies that the Fano plane is self-dual.

The specific representation, on the left, of the Levi graph of the Möbius–Kantor configuration (example 4 above) illustrates that a rotation of {{math|{{pi}}/4}} about the center (either clockwise or counterclockwise) of the diagram interchanges the blue and red vertices and maps edges to edges. That is to say that there exists a color interchanging automorphism of this graph. Consequently, the incidence structure known as the Möbius–Kantor configuration is self-dual.