It's a mystree

The minimum spanning tree of the graph is draw on the graph using bold purple lines. As you can see , the weights of edges $x$ , $y$ and $z$ are unknown.

The tightest bounds for $x$ , $y$ and $z$ can be written as below for some $p,q,r \in \mathbb{Z}^{+}$ :

$x \geq p$ $y \leq q$ $z \geq r$

What is the value of $p+q+r$ ?

Image Credit: papermadeofstone.com