Tree-optimized labeled directed graphs

Alexandru Chirvasitu ()
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Alexandru Chirvasitu: University at Buffalo

Journal of Combinatorial Optimization, 2023, vol. 45, issue 4, No 12, 10 pages

Abstract: Abstract For an additive submonoid $${\mathcal {M}}$$ M of $$\mathbb {R}_{\ge 0}$$ R ≥ 0 , the weight of a finite $${\mathcal {M}}$$ M -labeled directed graph is the sum of all of its edge labels, while the content is the product of the labels. Having fixed $${\mathcal {M}}$$ M and a directed tree E, we prove a general result on the shape of finite, acyclic, $${\mathcal {M}}$$ M -labeled directed graphs $$\Gamma $$ Γ of weight $$N\in {\mathcal {M}}$$ N ∈ M maximizing the sum of the contents of all copies $$E\subset \Gamma $$ 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 $$A^k$$ A k for a nilpotent $$\mathbb {R}_{\ge 0}$$ 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 $$\alpha $$ α arms in a directed graph with N edges.

Keywords: Acyclic directed graph; Labeled graph; Path; Star; 05C35; 05C20; 05C30 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10878-023-01022-9

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