Pappus graph: Difference between revisions
m dab. I'm not convinced the non-rectilinear crossing number is known for this graph, but the OEIS ref appears to be rectilinear only. |
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In the [[mathematics|mathematical]] field of [[graph theory]], the '''Pappus graph''' is a 3-[[regular graph|regular]] [[undirected graph]] with 18 vertices and 27 edges, formed as the [[Levi graph]] of the [[Pappus configuration]]<ref>{{MathWorld|urlname=PappusGraph|title=Pappus Graph}}</ref>. All the [[cubic graph|cubic]] [[distance-regular graph]]s are known.<ref>Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.</ref> The Pappus graph is one of the 13 such graphs. |
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Pappus graph''' is a 3-[[regular graph|regular]] [[undirected graph]] with 18 vertices and 27 edges, formed as the [[Levi graph]] of the [[Pappus configuration]]<ref>{{MathWorld|urlname=PappusGraph|title=Pappus Graph}}</ref>. All the [[cubic graph|cubic]] [[distance-regular graph]]s are known.<ref>Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.</ref> The Pappus graph is one of the 13 such graphs. |
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The Pappus graph has [[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]]. |
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]]. |
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The name "Pappus graph" has also been used to refer to a related nine-vertex graph<ref>{{citation |
The name "Pappus graph" has also been used to refer to a related nine-vertex graph<ref>{{citation |
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Revision as of 07:27, 28 March 2010
| 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 | Symmetric Distance-transitive Distance-regular Cubic Hamiltonian |
| Table of graphs and parameters | |
In the mathematical field of graph theory, the Pappus graph is a 3-regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration[1]. All the cubic distance-regular graphs are known.[2] 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 A110507 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 name "Pappus graph" has also been used to refer to a related nine-vertex graph[3], 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.[4][5]
The characteristic polynomial of the Pappus graph is . It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
Gallery
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Pappus graph coloured to highlight various cycles. -
The chromatic index of the Pappus graph is 3. -
The chromatic number of the Pappus graph is 2.
References
- ^ Weisstein, Eric W. "Pappus Graph". MathWorld.
- ^ Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
- ^ Kagno, I. N. (1947), "Desargues' and Pappus' graphs and their groups", American Journal of Mathematics, 69 (4): 859–863, doi:10.2307/2371806
- ^ Royle, G. "Cubic Symmetric Graphs (The Foster Census)."
- ^ Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.