On the irregularity strength of trees

For any graph G, let ni be the number of vertices of degree i, and λ(G) = maxi≤j{ ni+···+nj+i−1 j }. This is a general lower bound on the irregularity strength of graph G. All known facts suggest that for connected graphs, this is the actual irregularity strength up to an additive constant. In fact, this was conjectured to be the truth for regular graphs and for trees. Here we find an infinite sequence of trees with λ(T ) = n1 but strength converging to 11− √ 5 8 n1.