Triangle graph

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 is a particular finite undirected graph defined as follows:

It is a graph whose geometric realization is a triangle: it has three vertices and an edge between each pair of vertices.
It is the cycle graph on 3 vertices, and is denoted $C_{3}$ .
It is the complete graph on 3 vertices, and is denoted $K_{3}$ .
It is the odd graph $O_{2}$ , i.e., the Kneser graph $KG_{3,1}$ .

Arithmetic functions

Size measures

Function	Value	Explanation
size of vertex set	3	As $C_{n},n=3$ : $n=3$ As $K_{n},n=3$ : $n=3$ As $O_{n},n=2$ : ${\binom {2n-1}{n-1}}={\binom {3}{1}}=3$
size of edge set	3	As $C_{n},n=3$ : $n=3$ As $K_{n},n=3$ : ${\binom {n}{2}}={\binom {3}{2}}=3$ As $O_{n},n=2$ : ${\binom {2n-1}{n-1,n-1,1}}={\binom {3}{1,1,1}}=3$

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 $C_{n},n=3$ : 2 (independent of $n$ ) As $K_{n},n=3$ : $n-1=2$ As $O_{n},n=2$ : $n=2$
eccentricity of a vertex	1	As $C_{n},n=3$ : greatest integer function of $n/2$ equals greatest integer function of 3/2 equals 1 As $K_{n},n=3$ : 1 (true for any $n\geq 2$ ) As $O_{n},n=2$ : $n-1=1$

Other numerical invariants

Function	Value	Explanation
clique number	3	As $C_{n},n=3$ : 3 (since $n=3$ ; it's 2 for larger $n$ ) As $K_{n},n=3$ : $n=3$ As $O_{n},n=2$ : 3
independence number	1	As $C_{n},n=3$ : greatest integer function of $n/2$ equals greatest integer function of 3/2 equals 1 As $K_{n},n=3$ : 1 (independent of $n$ )
chromatic number	3	As $C_{n},n=3$ : 3 since $n$ is odd As $K_{n},n=3$ : $n=3$
radius of a graph	1	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
diameter of a graph	1	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
odd girth	3	As $C_{n},n=3$ : $n=3$ As $K_{n},n=3$ : 3 (true for any $n\geq 3$ ) As $O_{n},n=2$ : $2n-1=2(2)-1=3$
even girth	infinite	there are no cycles of even length
girth of a graph	3	minimum of even and odd girth