Global Topological Invariants of Stable Maps from 3-Manifolds to $\mathbb R^3$

With any stable map from a 3-manifold to $\mathbb R^3$, we associate a graph with weights in its vertices and edges. These graphs are $\mathcal A$-invariants from a global viewpoint. We study their properties and show that any tree with zero weights in its vertices and aleatory weights in its edges can be the graph of a stable map from $S^3$ to $\mathbb R^3$.