Solvsolitons associated with graphs

Abstract

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 solvsolitons). We classify them up to isometry, obtaining families depending on many parameters of explicit examples of Ricci solitons (of arbitrarily large dimension). A classification of graphs with up to 3 coherent components according to positivity is also given.