View source for Pappus graph

{{short description|Bipartite, 3-regular undirected graph}}
{{infobox graph
 | name = Pappus graph
 | image = [[Image:Pappus graph LS.svg|250px]]
 | image_caption = The Pappus graph.
 | namesake = [[Pappus of Alexandria]]
 | vertices = 18
 | edges = 27
 | automorphisms=216
 | girth = 6
 | radius = 4
 | diameter = 4
 | chromatic_number = 2
 | chromatic_index = 3
 | properties = [[Bipartite graph|Bipartite]]<br>[[Symmetric graph|Symmetric]]<br>[[Distance-transitive graph|Distance-transitive]]<br>[[Distance-regular graph|Distance-regular]]<br>[[Cubic graph|Cubic]]<br>[[Hamiltonian graph|Hamiltonian]]
|book thickness=3|queue number=2}}

In the [[mathematics|mathematical]] field of [[graph theory]], the '''Pappus graph''' is a [[Bipartite graph|bipartite]], 3-[[regular graph|regular]], [[undirected graph]] with 18 [[Vertex (graph theory)|vertices]] and 27 edges, formed as the [[Levi graph]] of the [[Pappus configuration]].<ref>{{MathWorld|urlname=PappusGraph|title=Pappus Graph}}</ref> It is named after [[Pappus of Alexandria]], an ancient [[Greek mathematics|Greek mathematician]] who is believed to have discovered the "hexagon theorem" describing the Pappus configuration. All the [[cubic graph|cubic]], [[distance-regular graph]]s are known; the Pappus graph is one of the 13 such graphs.<ref>Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.</ref>

The Pappus graph has [[Crossing number (graph theory)|rectilinear crossing number]] 5, and is the smallest cubic graph with that crossing number {{OEIS|id=A110507}}. It has [[Girth (graph theory)|girth]] 6, diameter 4, radius 4, [[chromatic number]] 2, [[chromatic index]] 3 and is both 3-[[k-vertex-connected graph|vertex-connected]] and 3-[[k-edge-connected graph|edge-connected]]. It has [[book thickness]] 3 and [[queue number]] 2.<ref>Jessica Wolz, ''Engineering Linear Layouts with SAT.'' Master Thesis, University of Tübingen, 2018</ref>

The Pappus graph has a [[chromatic polynomial]] equal to: 
<math>\begin{align}
(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 - 6609926x^5 \\
&+ 7238770x^4 - 6236975x^3 \\
&+ 3989074x^2 - 1690406x + 356509)
\end{align}</math>

The name "Pappus graph" has also been used to refer to a related nine-vertex graph,<ref>{{citation
  | last = Kagno | first = I. N.
  | title = Desargues' and Pappus' graphs and their groups
  | journal = American Journal of Mathematics
  | year = 1947
  | volume = 69
  | issue = 4
  | pages = 859–863
  | doi = 10.2307/2371806
  | jstor = 2371806
  | publisher = The Johns Hopkins University Press}}</ref> with a vertex for each point of the Pappus configuration and an edge for every pair of points on the same line; this nine-vertex graph is 6-regular, is the [[complement graph]] of the union of three disjoint [[triangle graph]]s, and is the complete tripartite graph K<sub>3,3,3</sub>. The first Pappus graph can be embedded in the torus to form a self-[[Petrie dual]] [[regular map (graph theory)|regular map]] with nine hexagonal faces; the second, to form a regular map with 18 triangular faces. The two regular toroidal maps are dual to each other.

==Algebraic properties==
The automorphism group of the Pappus graph is a group of order 216. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Pappus graph is a [[symmetric graph]]. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the ''Foster census'', the Pappus graph, referenced as F018A, is the only cubic symmetric graph on 18 vertices.<ref>Royle, G. [http://staffhome.ecm.uwa.edu.au/~00013890/remote/foster/ "Cubic Symmetric Graphs (The Foster Census)."] {{Webarchive|url=https://web.archive.org/web/20150912020037/http://staffhome.ecm.uwa.edu.au/~00013890/remote/foster/ |date=2015-09-12 }}</ref><ref>[[Marston Conder|Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.</ref>

The [[characteristic polynomial]] of the Pappus graph is <math>(x-3) x^4 (x+3) (x^2-3)^6</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

==Gallery==
<gallery>
Image:Pappus graph colored.svg|Pappus graph coloured to highlight various cycles.
Image:Pappus graph 3color edge.svg|The [[chromatic index]] of the Pappus graph is&nbsp;3.
Image:Pappus graph 2COL.svg|The [[chromatic number]] of the Pappus graph is&nbsp;2.
Image:Pappus graph as regular map.png|The Pappus graph embedded in the torus, as a regular map with nine hexagonal faces.
Image:Pappus-graph-on-torus.png|The Pappus graph and associated map embedded in the torus.
</gallery>

== References ==
{{reflist}}

[[Category:Individual graphs]]
[[Category:Regular graphs]]