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{{infobox graph
| name = F26 graph
| image = [[File:F26A graph.svg|220px]]
| image_caption = The F26A graph is Hamiltonian.
| namesake =
| vertices = 26
| edges = 39
| automorphisms= 78
| girth = 6
| diameter = 5
| radius = 5
| chromatic_number = 2
| chromatic_index = 3
| properties = [[Cayley graph]]<br>[[Symmetric graph|Symmetric]]<br>[[Cubic graph|Cubic]]<br>[[Hamiltonian graph| Hamiltonian]]<ref name="Mathworld"/>
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''F26A graph''' is a [[symmetric graph|symmetric]] [[bipartite graph|bipartite]] [[cubic graph]] with 26 vertices and 39 edges.<ref name="Mathworld">{{MathWorld|urlname=CubicSymmetricGraph |title=Cubic Symmetric Graph}}</ref>
 
It has [[chromatic number]]&nbsp;2, [[chromatic index]]&nbsp;3, [[graph diameter|diameter]]&nbsp;5, radius 5 and [[girth (graph theory)|girth]]&nbsp;6.<ref name="COND">[[Marston Conder|Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41&ndash;63, 2002.</ref> It is also a 3-[[k-vertex-connected graph|vertex-connected]] and 3-[[k-edge-connected graph|edge-connected]] graph.
 
The F26A graph is [[hamiltonian graph|Hamiltonian]] and can be described by the [[LCF notation]]&nbsp;[&minus;7,&nbsp;7]<sup>13</sup>.
 
==Algebraic properties==
The [[graph automorphism|automorphism group]] of the F26A graph is a group of order 78.<ref>Royle, G. [http://school.maths.uwa.edu.au/~gordon/remote/foster/F026A.html F026A data]</ref> It acts transitively on the vertices, on the edges, and on the arcs of the graph. Therefore the F26A graph is a [[symmetric graph]] (though not [[distance-transitive graph|distance transitive]]). It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the [[Foster census]], the F26A graph is the only cubic symmetric graph on 26 vertices.<ref name="COND"/> It is also a [[Cayley graph]] for the [[dihedral group]] ''D''<sub>26</sub>, generated by ''a'', ''ab'', and ''ab''<sup>4</sup>, where:<ref>[http://com2mac.postech.ac.kr/Lecture/Lec-7.pdf Yan-Quan Feng and Jin Ho Kwak, ''Cubic s-Regular Graphs'', p. 67.]</ref>
 
: <math>D_{26} = \langle a, b | a^2 = b^{13} = 1, aba = b^{-1} \rangle .</math>
 
The F26A graph is the smallest cubic graph where the automorphism group [[regular group action|acts regularly]] on arcs (that is, on edges considered as having a direction).<ref>Yan-Quan Feng and Jin Ho Kwak, "One-regular cubic graphs of order a small number times a prime or a prime square," ''J. Aust. Math. Soc.'' 76 (2004), 345-356 [http://www.austms.org.au/Publ/JAustMS/V76P3/y16.html].</ref>
 
The [[characteristic polynomial]] of the F26A graph is equal to
 
: <math>(x-3)(x+3)(x^4-5x^2+3)^6. \, </math>
 
==Gallery==
<gallery>
Image:F26A graph 2COL.svg |The [[chromatic number]] of the F26A graph is&nbsp;2.
Image:F26A graph 3color edge.svg|The [[chromatic index]] of the F26A graph is&nbsp;3.
Image:F26A graph alt.svg|Alternative drawing of the F26A graph.
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Latest revision as of 12:59, 17 September 2013

Template:Infobox graph In the mathematical field of graph theory, the F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges.[1]

It has chromatic number 2, chromatic index 3, diameter 5, radius 5 and girth 6.[2] It is also a 3-vertex-connected and 3-edge-connected graph.

The F26A graph is Hamiltonian and can be described by the LCF notation [−7, 7]13.

Algebraic properties

The automorphism group of the F26A graph is a group of order 78.[3] It acts transitively on the vertices, on the edges, and on the arcs of the graph. Therefore the F26A graph is a symmetric graph (though not distance transitive). It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the F26A graph is the only cubic symmetric graph on 26 vertices.[2] It is also a Cayley graph for the dihedral group D26, generated by a, ab, and ab4, where:[4]

The F26A graph is the smallest cubic graph where the automorphism group acts regularly on arcs (that is, on edges considered as having a direction).[5]

The characteristic polynomial of the F26A graph is equal to

Gallery

References

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  1. ↑

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  2. ↑ 2.0 2.1 Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41–63, 2002.
  3. ↑ Royle, G. F026A data
  4. ↑ Yan-Quan Feng and Jin Ho Kwak, Cubic s-Regular Graphs, p. 67.
  5. ↑ Yan-Quan Feng and Jin Ho Kwak, "One-regular cubic graphs of order a small number times a prime or a prime square," J. Aust. Math. Soc. 76 (2004), 345-356 [1].
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