Pappus graph

Pappus graph
The Pappus graph.
Named after	Pappus of Alexandria
Vertices	18
Edges	27
Radius	4
Diameter	4
Girth	6
Automorphisms	216
Chromatic number	2
Chromatic index	3
Properties	Bipartite Symmetric Distance-transitive Distance-regular Cubic Hamiltonian

In the mathematical field of graph theory, the Pappus graph is a bipartite 3-regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the "hexagon theorem" describing the Pappus configuration. All the cubic distance-regular graphs are known; the Pappus graph is one of the 13 such graphs.

The Pappus graph has rectilinear crossing number 5, and is the smallest cubic graph with that crossing number (sequence in the OEIS). It has girth 6, diameter 4, radius 4, chromatic number 2, chromatic index 3 and is both 3-vertex-connected and 3-edge-connected.

The Pappus graph has a chromatic polynomial equal to: $(x-1)x(x^{16}-26x^{15}+325x^{14}-2600x^{13}+14950x^{12}-65762x^{11}+$ $(x-1)x(x^{{16}}-26x^{{15}}+325x^{{14}}-2600x^{{13}}+14950x^{{12}}-65762x^{{11}}+$ $229852x^{10}-653966x^{9}+1537363x^{8}-3008720x^{7}+4904386x^{6}-$ $229852x^{{10}}-653966x^{9}+1537363x^{8}-3008720x^{7}+4904386x^{6}-$ $6609926x^{5}+7238770x^{4}-6236975x^{3}+3989074x^{2}-1690406x+356509)$ $6609926x^{5}+7238770x^{4}-6236975x^{3}+3989074x^{2}-1690406x+356509)$ .

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