(k,g)-Cage — различия между версиями
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| − | '''<math>(k,g)</math>-Cage''' - | + | '''<math>\,(k,g)</math>-Cage''' — ''[[(k,g)-Клетка|<math>\,(k,g)</math>-клетка]].'' |
| − | For a given ordered pair of integers <math>(k,g)</math>, with <math>k \geq 2</math> and <math>g | + | For a given ordered pair of integers <math>\,(k,g)</math>, with <math>k \geq 2</math> and <math>g |
| − | \geq 3</math>, a <math>k</math>-regular graph with the smallest cycle length, or girth, | + | \geq 3</math>, a <math>\,k</math>-[[regular graph]] with the smallest [[cycle]] length, or girth, |
| − | equal to <math>g</math> is said to be a <math>(k,g)</math>-graph. A '''<math>(k,g)</math>-cage''' is a | + | equal to <math>\,g</math> is said to be a <math>\,(k,g)</math>-graph. A '''<math>\,(k,g)</math>-cage''' is a |
| − | <math>(k,g)</math>-graph having the least number, <math>f(k,g)</math>, of vertices. We call | + | <math>\,(k,g)</math>-graph having the least number, <math>\,f(k,g)</math>, of [[vertex|vertices]]. We call |
| − | <math>f(k,g)</math> the ''' cage number''' of a <math>(k,g)</math>-graph. One readily | + | <math>\,f(k,g)</math> the '''[[cage number]]''' of a <math>\,(k,g)</math>-graph. One readily |
| − | observes that <math>(2,g)</math>-cages are cycles of length <math>g</math>, and | + | observes that <math>\,(2,g)</math>-cages are cycles of length <math>\,g</math>, and |
| − | <math>(k,3)</math>-cages are complete graphs of order <math>k+1</math>. | + | <math>\,(k,3)</math>-cages are [[complete graph|complete graphs]] of order <math>\,k+1</math>. |
| − | The unique (3,7)-cage known as the '''McGee graph''' is an example of | + | The unique <math>\,(3,7)</math>-cage known as the '''[[McGee graph]]''' is an example of |
| − | a cage that is not transitive. It has 24 vertices and its automorphism | + | a cage that is not transitive. It has <math>\,24</math> vertices and its automorphism group has order <math>\,32</math>. |
| − | group has order 32. | + | |
| + | ==Литература== | ||
| + | |||
| + | * Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009. | ||
Текущая версия на 11:50, 24 апреля 2012
-Cage — -клетка.
For a given ordered pair of integers , with 
and 
, a 
-regular graph with the smallest cycle length, or girth,
equal to 
is said to be a -graph. A -cage is a
-graph having the least number, 
, of vertices. We call
the cage number of a -graph. One readily
observes that 
-cages are cycles of length , and
-cages are complete graphs of order 
.
The unique 
-cage known as the McGee graph is an example of
a cage that is not transitive. It has 
vertices and its automorphism group has order 
.
Литература
- Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.
