On the existence of nilsolitons on 2-step nilpotent Lie groups

A 2-step nilpotent Lie algebra n is said to be of type (p,q)if dim(n)=p+q and dim([n,n])=p. By considering a class of 2-step nilpotent Lie algebras naturally attached to graphs, we prove that there exist indecomposable, 2-step nilpotent Lie groups of type (p,q) which do not admit a nilsoliton metric for every pair (p,q) such that 20 < q and q-2 < p < 1/2q^2-5/2q+10. This improves a result due to Jablonski.