Tree-optimized labeled directed graphs

Abstract

For an additive submonoid ${\mathcal {M}}$ of $\mathbb {R}_{\ge 0}$, the weight of a finite ${\mathcal {M}}$-labeled directed graph is the sum of all of its edge labels, while the content is the product of the labels. Having fixed ${\mathcal {M}}$ and a directed tree E, we prove a general result on the shape of finite, acyclic, ${\mathcal {M}}$-labeled directed graphs $\Gamma$ of weight $N\in {\mathcal {M}}$ maximizing the sum of the contents of all copies $E\subset \Gamma$. 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 $\mathbb {R}_{\ge 0}$-valued square matrix A whose entries add up to N. Finally, we apply the same techniques to obtain the maximal number of stars with $\alpha$ arms in a directed graph with N edges.