Does the notion of graphs with vertex multiplicity exist? 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 ?
 A: In this paper, they call these objects "blow-up graphs", since the operation of adding twins to a vertex is commonly called a blow-up. Putting integer weights on the vertices of a graph to signify by how much to blow up a vertex like you do above is fairly common. 
A: This notion is complementary to the notion of vertex multiplication, in which every vertex is replaced with a homogeneous clique.  This goes back to Lovasz' proof of the Weak Perfect Graph Theorem, and probably even earlier.
A: I have heard this concept called, logically enough, "vertex multiplication."  The first hit I got when I searched was Small Survey on Perfect Graphs by Michele Alberti.  Be aware, though, that I've also heard the term "vertex multiplication" used when the multiple copies of a vertex form a clique rather than an independent set.
A: As well as the use of this by Lovász for perfect graphs (mentioned by earlier answers), this has been used by Häggkvist to find high-degree four-chromatic triangle-free graphs. See
Häggkvist, R. (1981), "Odd cycles of specified length in nonbipartite graphs", Graph Theory (Cambridge, 1981), pp. 89–99, MR0671908.
A: *

*You might want to look up this paper where reduced graphs are studied, in the sense of Servatius (or very similar to it), from the viewpoint of spectral graph theory.

*People have studied twin-free graphs quite a lot. One possible pointer is this.
A: The equivalence relation that HJRW defines in the comments (that is twin vertices may be adjacent) is related to the cochromatic number $z(G)$ of a graph $G$.  The cochromatic number of $G$ is the minimum number of colours needed to colour $V(G)$ such that each colour class induces a clique or stable set.  Thus, the number of such equivalence classes gives an upper bound on the cochromatic number.  
A: A graph in which no two vertices have the same neighborhood is called a point-determining graph or mating graph or mating-type graph. See  A006024 for references.
