Undergrad Spin networks: what exactly is a trivalent node?

[Heidi]

Hi Pfs
Rovelli defines spin networks in this paper
https://arxiv.org/abs/1004.1780
for a trivalent node Vn = 0 (the volume)
nodes begin to "get" volume with the four valent case.
take a cube or a tetrahedron, each vertex is linkes to 3 nodes so they would have a null volume.
things are different if we take the reciprocal network: from a node inside a tetrahedron four edges can intersect the four faces.
then we are in the four valent case.
have we to be more precise to talk about the valence of a node?

 

[Delta Prime]

Rovelli defines spin networks in this paper

Graph theory nomenclature can be confusing, definitions must often be given to maintain the proper content.
And believe it or not some definitions are even excluded from certain theorems.
What is a valence of a node, trivalent node.
A trivalent (3-valent) graph is often called a cubic graph.
A graph consists of a set N of items called nodes (vertices, points etc.), with a set E ⊆ the set of (unordered) pairs of nodes.
The pairs that are in E are called edges (links, arcs etc.) & an edge is said to
“run between” its two nodes, its “endpoints”.