Counting paths in directed graphs
Volume 73 / 2025
Bulletin Polish Acad. Sci. Math. 73 (2025), 99-109
MSC: Primary 05C20; Secondary 05C30
DOI: 10.4064/ba250507-22-12
Published online: 20 March 2026
Abstract
We consider the class of directed graphs with $N\geq 1$ edges and without loops shorter than $k\ge 1$. Using the concept of a labelled graph, we determine graphs from this class that maximize the number of 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 $N\geq 1$ edges.
