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| edges = 27
| edges = 27
| automorphisms=216
| automorphisms=216
| chromatic_number =
| girth = 6
| diameter = 4
| chromatic_number = 2
| chromatic_index =
| chromatic_index =
| properties = [[Distance-regular graph|Distance-regular]]<br/>[[Cubic graph|Cubic]]
| properties = [[Symmetric graph|Symmetric]]<br>[[Distance-regular graph|Distance-regular]]<br>[[Cubic graph|Cubic]]<br>[[Hamiltonian graph|Hamiltonian]]
}}
}}


In the [[mathematics|mathematical]] field of [[graph theory]], the '''Pappus graph''' is a 3-[[regular graph]] with 18 vertices and 27 edges, formed as the [[Levi graph]] of the [[Pappus configuration]]. It is a [[distance-regular graph]], one of only 14 such [[cubic graph]]s according to the [[Foster census]]. It has [[crossing number]] 5, and is the smallest cubic graph with that crossing number {{OEIS|id=A110507}}. Its automorphism group has order 216.
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Pappus graph''' is a 3-[[regular graph]] with 18 vertices and 27 edges, formed as the [[Levi graph]] of the [[Pappus configuration]]<ref>{{MathWorld|urlname=PappusGraph|title=Pappus Graph}}</ref>. It is a [[distance-regular graph]], one of only 14 such [[cubic graph]]s according to the ''[[Foster census]]''. It has [[crossing number]] 5, and is the smallest cubic graph with that crossing number {{OEIS|id=A110507}}.


The name "Pappus graph" has also been used to refer to a related nine-vertex graph (Kagno 1947), with a point for each point of the Pappus configuration and an edge for every pair of points on the same line.
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}}</ref>, with a point for each point of the Pappus configuration and an edge for every pair of points on the same line.

==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://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."]</ref><ref>Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.</ref>


==Gallery==
==Gallery==
Line 24: Line 37:


== References ==
== References ==
{{reflist}}

* {{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}}

== External links ==

* {{MathWorld|urlname=PappusGraph|title=Pappus Graph}}


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

Revision as of 07:12, 2 July 2009

Pappus graph
The Pappus graph.
Named afterPappus of Alexandria
Vertices18
Edges27
Diameter4
Girth6
Automorphisms216
Chromatic number2
PropertiesSymmetric
Distance-regular
Cubic
Hamiltonian
Table of graphs and parameters

In the mathematical field of graph theory, the Pappus graph is a 3-regular graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration[1]. It is a distance-regular graph, one of only 14 such cubic graphs according to the Foster census. It has crossing number 5, and is the smallest cubic graph with that crossing number (sequence A110507 in the OEIS).

The name "Pappus graph" has also been used to refer to a related nine-vertex graph[2], with a point for each point of the Pappus configuration and an edge for every pair of points on the same line.

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.[3][4]

  • Pappus graph colored to highlight various cycles.
    Pappus graph colored to highlight various cycles.
  • The edges of the Pappus graph can be colored using 3 colors.
    The edges of the Pappus graph can be colored using 3 colors.
  • The chromatic number of the Pappus graph is 2.
    The chromatic number of the Pappus graph is 2.

References

  1. ^ Weisstein, Eric W. "Pappus Graph". MathWorld.
  2. ^ Kagno, I. N. (1947), "Desargues' and Pappus' graphs and their groups", American Journal of Mathematics, 69 (4): 859–863, doi:10.2307/2371806
  3. ^ Royle, G. "Cubic Symmetric Graphs (The Foster Census)."
  4. ^ Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.
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