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), but the term has been extended by H.S.M. Coxeter to refer to an incidence graph of any incidence structure. Levi graph examples The Levi graph of the Fano plane is the Heawood graph. Since the Heawood graph is 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 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. Generalization It is possible to generalize the notion of an incidence structure to include more than two types of objects. A structure with types of objects is called an incidence structure of rank or a rank geometry. Formally these are defined as tuples with and The Levi graph for these structures is defined as a multipartite graph with vertices corresponding to each type being colored the same. See also Incidence (geometry) Incidence geometry Projective configuration Abstract polytope Notes References G. Eric Moorhouse (2014) Incidence Geometry via John Baez at University of California, Riverside Further reading CRC Press (2000). Handbook of discrete and combinatorial mathematics, (Chapter 12.2), Harold L. Dorwart (1966) The Geometry of Incidence, Prentice Hall Families of sets Combinatorics Finite geometry Incidence geometry