what is mean "Number of connected Triplets of vertices" in global clustering I am really confuse to understand what is mean "Number of connected Triplets of vertices" in global.
The global clustering is defines by:
c= 3 * Number of Triangle/Number of connected Triplets of vertices
I could not understand what is exactly Triples of vertices.If I get any help it would really appreciated.  
 A: One very concrete way to define the global clustering coefficient of a graph with vertex set $V$ and edge set $E$ is
$$
  C_{GC} = \frac{\text{# of ordered triples $(x,y,z) \in V^3$ such that $xy, yz, xz \in E$}}{\text{# of ordered triples $(x,y,z) \in V^3$ such that $xy, yz \in E$}}.
$$
Another, equivalent definition (the one you're referring to) is
$$
  C_{GC} = \frac{\text{# of edge pairs $\{xy, yz\} \subset E$, with $x\ne z$, such that $xz \in E$}}{\text{# of edge pairs $\{xy, yz\} \subset E$, with $x\ne z$}}.
$$
In the second definition, both the numerator and denominator will be exactly $\frac12$ of what they are in the first definition (since an edge pair $\{xy,yz\}$ can correspond to both the ordered triple $(x,y,z)$ and the ordered triple $(z,y,x)$). This doesn't change their ratio, so we get the same value of $C_{GC}$ either way.
A "connected triplet" is the denominator of the second definition: a pair of edges $xy, yz$ sharing one endpoint $y$. You could equivalently define a connected triplet as a subgraph isomorphic to $P_3$ (a path on $3$ vertices).
The numerator of the second definition is $3$ times the number of triangles if we define a triangle as a subgraph isomorphic to $C_3$ (a cycle on $3$ vertices), or a set of three vertices $\{x,y,z\}$ with all three edges $xy, yz, xz$ present. We have to multiply by $3$ because we can think of each such triangle in three ways:


*

*An edge pair $\{xy,yz\}$ such that $xz \in E$,

*An edge pair $\{xz,zy\}$ such that $xy \in E$,

*An edge pair $\{yx,xz\}$ such that $yz \in E$.


As a result, the numerator of the second definition counts each triangle three times.
