Mass–luminosity relation: Difference between revisions

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{{infobox graph
| name = Meredith graph
| image = [[Image:Meredith graph.svg|220px]]
| image_caption = The Meredith graph
  | namesake = G. H. Meredith
| vertices = 70
| edges = 140
| automorphisms =
| girth = 4
| diameter = 8
| radius = 7
| chromatic_number = 3
| chromatic_index = 5
| properties = [[Eulerian graph|Eulerian]]
}}
 
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Meredith graph''' is a 4-[[regular graph|regular]] [[undirected graph]] with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973.<ref>{{MathWorld|urlname=MeredithGraph|title=Meredith graph}}</ref>  
 
The Meredith graph is 4-[[k-vertex-connected graph|vertex-connected]] and 4-[[k-edge-connected graph|edge-connected]], has [[chromatic number]] 3, [[chromatic index]] 5, radius 7, diameter 8, girth 4 and is [[Hamiltonian graph|non-hamiltonian]].<ref>Bondy, J. A. and Murty, U. S. R. "Graph Theory". Springer, p. 470, 2007.</ref>
 
Published in 1973, it provides a counterexample to the [[Crispin Nash-Williams]] conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian.<ref>Meredith, G. H. J. "Regular 4-Valent 4-Connected Nonhamiltonian Non-4-Edge-Colorable Graphs." J. Combin. Th. B 14, 55-60, 1973.</ref><ref>Bondy, J. A. and Murty, U. S. R. "Graph Theory with Applications". New York: North Holland, p. 239, 1976.</ref> However, [[W. T. Tutte]] showed that all 4-connected [[planar graph]]s are hamiltonian.<ref>Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969.</ref>
 
The [[characteristic polynomial]] of the Meredith graph is <math>(x-4) (x-1)^{10} x^{21} (x+1)^{11} (x+3) (x^2-13) (x^6-26 x^4+3 x^3+169 x^2-39 x-45)^4</math>.
 
==Gallery==
<gallery>
Image:Meredith graph 3COL.svg|The [[chromatic number]] of the Meredith graph is&nbsp;3.
Image:Meredith graph 5color edge.svg|The [[chromatic index]] of the Meredith graph is&nbsp;5.
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Revision as of 01:13, 14 December 2013

Template:Infobox graph

In the mathematical field of graph theory, the Meredith graph is a 4-regular undirected graph with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973.[1]

The Meredith graph is 4-vertex-connected and 4-edge-connected, has chromatic number 3, chromatic index 5, radius 7, diameter 8, girth 4 and is non-hamiltonian.[2]

Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian.[3][4] However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.[5]

The characteristic polynomial of the Meredith graph is .

Gallery

References

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  2. ↑ Bondy, J. A. and Murty, U. S. R. "Graph Theory". Springer, p. 470, 2007.
  3. ↑ Meredith, G. H. J. "Regular 4-Valent 4-Connected Nonhamiltonian Non-4-Edge-Colorable Graphs." J. Combin. Th. B 14, 55-60, 1973.
  4. ↑ Bondy, J. A. and Murty, U. S. R. "Graph Theory with Applications". New York: North Holland, p. 239, 1976.
  5. ↑ Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969.
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