I need to use graphs where each vertex gets a natural number, $b(v)$, its multiplicity. These numbers indicate how many 'replications' of the vertex we have.

It is actually a way to write in a compact way a graph which has a lot of twin vertices: two vertices $u$ and $v$ are twin if $N(u)=N(v)$ (they are not adjacent).

Since the twin relation is an equivalence relation, there is a unique way to write any graph in a reduced form as a graph with vertex multiplicities.

Here is an example:

This representation is particularly useful when vertices represent some items with large quantity.

Does someone know if such a concept already exists ? What's the usual name ? Do you have some references ?

J. Algebra126 (1989), no. 1, 34–60. – HJRW Sep 18 '13 at 9:49Context Orbifolds, Diploma Thesis, TU Dresden, 2009, link). – Tobias Schlemmer Sep 18 '13 at 22:22