Desargues graph: Difference between revisions

Latest revision as of 22:18, 29 May 2012

This article defines a particular undirected graph, i.e., the definition here determines the graph uniquely up to graph isomorphism.
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Definition

The Desargues graph is a particular graph on 20 vertices defined in the following equivalent ways:

It is the Levi graph of the Desargues configuration
It is the bipartite Kneser graph with parameters 5,2
It is the generalized Petersen graph $G (10, 3)$ .

Arithmetic functions

Size measures

Function	Value	Explanation
size of vertex set	20	As Levi graph of Desargues configuration: number of points + number of lines in the configuration = $10 + 10 = 20$ As bipartite Kneser graph with parameters $n = 5, k = 2$ : $2 (\binom{n}{k}) = 2 (\binom{5}{2}) = 2 (10) = 20$ As generalized Petersen graph $G (n, k), n = 10, k = 3$ : $2 (10) = 20$
size of edge set	30	As Levi graph of Desargues configuration: number of incidences between points and lines = (number of lines) * (number of points on each line) = $(10) (3) = 30$ As bipartite Kneser graph with parameters $n = 5, k = 2$ : $(\binom{n}{k, k, n - 2 k}) = \frac{n!}{k! k! (n - 2 k)!} = (\binom{5}{2, 2, 1}) = 30$ As generalized Petersen graph $G (n, k), n = 10, k = 3$ : $3 n = 3 (10) = 30$

@@ Line 8: / Line 8: @@
 # It is the [[defining ingredient::bipartite Kneser graph]] with parameters 5,2
 # It is the [[defining ingredient::generalized Petersen graph]] <math>G(10,3)</math>.
+==Arithmetic functions==
+===Size measures===
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| {{arithmetic function value|size of vertex set|20}} || As Levi graph of Desargues configuration: number of points + number of lines in the configuration = <math>10 + 10 = 20</math><br>As bipartite Kneser graph with parameters <math>n = 5, k = 2</math>: <math>2 \binom{n}{k} = 2 \binom{5}{2} = 2(10) = 20</math><br>As generalized Petersen graph <math>G(n,k), n = 10, k = 3</math>: <math>2(10) = 20</math>
+|-
+| {{arithmetic function value|size of edge set|30}} || As Levi graph of Desargues configuration: number of incidences between points and lines = (number of lines) * (number of points on each line) = <math>(10)(3) = 30</math><br>As bipartite Kneser graph with parameters <math>n = 5, k = 2</math>: <math>\binom{n}{k,k,n-2k} =\frac{n!}{k!k! (n - 2k)!} = \binom{5}{2,2,1} = 30</math><br>As generalized Petersen graph <math>G(n,k), n = 10, k = 3</math>: <math>3n = 3(10) = 30</math>
+|}