# What does Rovelli mean with "oriented and ordered graph"?

• A
• Heidi
In summary, Rovelli discusses spin networks in his book Quantum Gravity. These are oriented and ordered graphs that can be transformed through diffeomorphisms. The links in a spin network have a source and target function, which determines their orientation. The order of the links is important, as it is related to the labeling and coloring of the links. However, it is not necessary for the links to be oriented or ordered in order to be colored. The spin networks were inspired by Penrose's diagrams, which were trivalent and had no explicit intertwiners. The links in these diagrams were not oriented and were colored by numbers. Rovelli also explains that changing the orientation of a link in a spin network will result in replacing

#### Heidi

Hi Pfs
Rovelli writes this in his book (Qunatum Gravity) about spin networks:
Given an oriented and ordered graph there is a finite disgrete group of maps that change its order or orientation and that can be obtained as a diffeomorphism.
A link is equipped the source and target functions. this give the orientation.
But what is the order he is talking about.
the paragraph is the 6.4 (Diff invariance)
thanks

You have to label the links first, as ##l_1, l_2, \dots## say, before you can assign a colouring ##j_1 , j_2 , \dots##. It is this labelling of the links that is the ordering, I think. With some graphs there are diffeomorphisms that simply swap the links around and hence change the ordering.

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Fractal matter
yes we can do like that to color the links but it is not necessary to have oriented ordered links.
The Penrose's diagrams were the ancestors of the spin networks.
they were trivalent with no explicit intertwiners. the links were not oriented and each link was coloured by a number (not a representation)
So we had a numerical function on a graph without necessary ordering:
to each pair of connected node we assign a number.

Rovelli writes later that if changing the order correspond to swap the variables, changing the orientation leads to replace a variable by its inverse.
If in an oriented loop the holonomy gives a matrix, changing the orientation gives the inverse matrix.

I think, ordering means specifying the predecessor and successor to each node of the graph.