  We show how to associate with each graph with a certain property (positivity)
a family of simply connected solvable Lie groups endowed with left-invariant
Riemannian metrics that are Ricci solitons (called solsolitons). We classify
them up to isometry, obtaining families depending on many parameters of
explicit examples of Ricci solitons. A classification of graphs with up to 3
coherent components according to positivity is also given.
