Cycle graph:C5: Difference between revisions

Revision as of 17:53, 28 May 2012

This article defines a particular undirected graph, i.e., the definition here determines the graph uniquely up to graph isomorphism.
View a complete list of particular undirected graphs

Definition

This undirected graph is defined in the following equivalent ways:

It is the cycle graph on 5 vertices, i.e., the graph $C_{5}$
It is the Paley graph $P_{5}$ corresponding to the field of 5 elements
It is the unique (up to graph isomorphism) self-complementary graph on a set of 5 vertices

Note that 5 is the only size for which the Paley graph coincides with the cycle graph. In general, the Paley graph $P_{q}$ can be expressed as an edge-disjoint union of $(q-1)/4$ cycle graphs. In our case, $(5-1)/4=1$ , so the graphs coincide.

Arithmetic functions

Size measures

Function	Value	Explanation
size of vertex set	5	By either definition
size of edge set	5	As $C_{n},n=5$ : $n=5$ As $P_{q},q=5$ : $q(q-1)/4=5(4)/4=5$

Numerical invariants associated with vertices

Since the graph is a vertex-transitive graph, any numerical invariant associated to a vertex must be equal on all vertices of the graph. Below are listed some of these invariants:

Function	Value	Explanation
degree of a vertex	2	As $C_{n},n=5(n\geq 3)$ : 2 As $P_{q},q=5$ : $(q-1)/2=(5-1)/2=2$
eccentricity of a vertex	2	As $C_{n},n=5(n\geq 3)$ : greatest integer of $n/2$ equals 2 As $P_{q},q=5$ : 2 (follows from every element of a finite field is expressible as a sum of two squares)

Other numerical invariants

Function	Value	Explanation
clique number	2	As cycle graph $C_{n},n=5$ : 2 (since $n\geq 4$ )
independence number	2	As cycle graph $C_{n},n=5$ : Greatest integer of $n/2$ , which is greatest integer of $5/2$ , which is 2
chromatic number	3	As cycle graph $C_{n},n=5$ : 3 since $n$ is odd
radius of a graph	2	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
diameter of a graph	2	Due to vertex-transitivity, the diameter equals the eccentricity of any vertex, which has been computed above.

Graph properties

Property	Satisfied?	Explanation
self-complementary graph	Yes	Paley graphs are self-complementary
strongly regular graph	Yes	Paley graphs are strongly regular
regular graph	Yes	Follows from being strongly regular. Also follows from the fact that cycle graphs are 2-regular.
conference graph	Yes	Paley graphs are conference graphs
symmetric graph	Yes
edge-transitive graph	Yes	Follows on account of being Paley and also on account of being cyclic
vertex-transitive graph	Yes

@@ Line 21: / Line 21: @@
 |-
 | {{arithmetic function value|size of edge set|5}} || As <matH>C_n, n = 5</math>: <math>n = 5</math><br>As <math>P_q, q = 5</math>: <math>q(q - 1)/4 = 5(4)/4 = 5</math>
+|}
+===Numerical invariants associated with vertices===
+Since the graph is a [[vertex-transitive graph]], any numerical invariant associated to a vertex must be equal on all vertices of the graph. Below are listed some of these invariants:
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| {{arithmetic function value|degree of a vertex|2}} || As <math>C_n, n = 5 (n \ge 3)</math>: 2<br>As <math>P_q, q = 5</math>: <math>(q -1)/2 = (5 - 1)/2 = 2</math>
+|-
+| {{arithmetic function value|eccentricity of a vertex|2}} || As <matH>C_n, n = 5 (n \ge 3)</math>: greatest integer of <matH>n/2</math> equals 2<br>As <math>P_q, q = 5</math>: 2 (follows from [[groupprops:every element of a finite field is expressible as a sum of two squares|every element of a finite field is expressible as a sum of two squares]])
 |}
@@ Line 33: / Line 45: @@
 |-
 | {{arithmetic function value|chromatic number|3}} || As cycle graph <math>C_n, n = 5</math>: 3 since <matH>n</math> is odd
+|-
+| {{arithmetic function value|radius of a graph|2}} || Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
+|-
+| {{arithmetic function value|diameter of a graph|2}} || Due to vertex-transitivity, the diameter equals the eccentricity of any vertex, which has been computed above.
 |}