Tree-optimized labeled directed graphs

For an additive submonoid M of R ≥ 0 , the weight of a finite M -labeled directed graph is the sum of all of its edge labels, while the content is the product of the labels. Having fixed M and a directed tree E , we prove a general result on the shape of finite, acyclic, M -labeled directed graphs Γ of weight N ∈ M maximizing the sum of the contents of all copies E ⊂ Γ . This specializes to recover a result of Hajac and the author’s on the maximal number of length- k paths in an acyclic directed graph with N edges. It also applies to prove a conjecture by the same authors on the maximal sum of entries of A k for a nilpotent R ≥ 0 -valued square matrix A whose entries add up to N . Finally, we apply the same techniques to obtain the maximal number of stars with α arms in a directed graph with N edges.