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{{infobox graph | |||
| name = Biggs–Smith graph | |||
| image = [[Image:Biggs-Smith graph.svg|250px]] | |||
| image_caption = The Biggs–Smith graph | |||
| namesake = | |||
| vertices = 102 | |||
| edges = 153 | |||
| automorphisms= 2448 ([[projective special linear group|PSL]](2,17)) | |||
| girth = 9 | |||
| radius = 7 | |||
| diameter = 7 | |||
| chromatic_number = 3 | |||
| chromatic_index = 3 | |||
| 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 '''Biggs–Smith graph''' is a 3-[[regular graph]] with 102 vertices and 153 edges.<ref>{{MathWorld|urlname=Biggs-SmithGraph|title=Biggs–Smith Graph}}</ref> | |||
It has chromatic number 3, chromatic index 3, radius 7, diameter 7 and girth 9. It is also a 3-[[k-vertex-connected graph|vertex-connected graph]] and a 3-[[k-edge-connected graph|edge-connected graph]]. | |||
All the [[cubic graph|cubic]] [[distance-regular graph]]s are known.<ref>[[A. E. Brouwer|Brouwer, A. E.]]; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.</ref> The Biggs–Smith graph is one of the 13 such graphs. | |||
==Algebraic properties== | |||
The automorphism group of the Biggs–Smith graph is a group of order 2448<ref>Royle, G. [http://www.csse.uwa.edu.au/~gordon/foster/F102A.html F102A data]</ref> isomorphic to the [[projective special linear group]] PSL(2,17). It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Biggs–Smith 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 Biggs–Smith graph, referenced as F102A, is the only cubic symmetric graph on 102 vertices.<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 Biggs–Smith graph is also uniquely determined by the its [[graph spectrum]], the set of graph eigenvalues of its [[adjacency matrix]].<ref>E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189–202, 2003</ref> | |||
The [[characteristic polynomial]] of the Biggs–Smith graph is : | |||
<math>(x-3) (x-2)^{18} x^{17} (x^2-x-4)^9 (x^3+3 x^2-3)^{16}</math>. | |||
==Gallery== | |||
<gallery> | |||
Image:Biggs-Smith graph 3COL.svg|The [[chromatic number]] of the Biggs–Smith graph is 3. | |||
Image:Biggs-Smith graph 3color edge.svg|The [[chromatic index]] of the Biggs–Smith graph is 3. | |||
Image:BiggsSmith.svg|Alternative drawing of the Biggs–Smith graph. | |||
Image:Biggs-Smith graph - circle-based drawing.jpg|Decomposition of the Biggs–Smith graph into 6 sets of size 17. | |||
</gallery> | |||
== References == | |||
{{reflist}} | |||
*On trivalent graphs, NL Biggs, DH Smith - Bulletin of the London Mathematical Society, 3 (1971) 155-158. | |||
{{DEFAULTSORT:Biggs-Smith graph}} | |||
[[Category:Individual graphs]] | |||
[[Category:Regular graphs]] | |||
Latest revision as of 17:17, 20 March 2013
In the mathematical field of graph theory, the Biggs–Smith graph is a 3-regular graph with 102 vertices and 153 edges.[1]
It has chromatic number 3, chromatic index 3, radius 7, diameter 7 and girth 9. It is also a 3-vertex-connected graph and a 3-edge-connected graph.
All the cubic distance-regular graphs are known.[2] The Biggs–Smith graph is one of the 13 such graphs.
Algebraic properties
The automorphism group of the Biggs–Smith graph is a group of order 2448[3] isomorphic to the projective special linear group PSL(2,17). It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Biggs–Smith 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 Biggs–Smith graph, referenced as F102A, is the only cubic symmetric graph on 102 vertices.[4]
The Biggs–Smith graph is also uniquely determined by the its graph spectrum, the set of graph eigenvalues of its adjacency matrix.[5]
The characteristic polynomial of the Biggs–Smith graph is : .
Gallery
-
The chromatic number of the Biggs–Smith graph is 3. -
The chromatic index of the Biggs–Smith graph is 3. -
Alternative drawing of the Biggs–Smith graph. -
Decomposition of the Biggs–Smith graph into 6 sets of size 17.
References
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- On trivalent graphs, NL Biggs, DH Smith - Bulletin of the London Mathematical Society, 3 (1971) 155-158.
- ↑
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Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog. - ↑ Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
- ↑ Royle, G. F102A data
- ↑ Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41–63, 2002.
- ↑ E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189–202, 2003