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arXiv · 2209.08944

Counting paths in directed graphs

Piotr M. Hajac·Oskar M. Stachowiak
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Abstract

We consider the class of directed graphs with $N\geq 1$ edges and without loops shorter than $k\geq1$. Using the concept of a labelled graph, we determine graphs from this class that maximize the number of all paths of length $k$. Then we show an $R$-labelled version of this result for semirings $R$ contained in the semiring of non-negative real numbers and containing the semiring of non-negative rational numbers. We end by posing a related open problem concerning the maximal dimension of the path algebra of a connected acyclic directed graph with $N\geq1$ edges.

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Piotr M. Hajac, Oskar M. Stachowiak. 2026-03-23. Counting paths in directed graphs. https://doi.org/10.4064/ba250507-22-12

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