Counting paths in directed graphs

TL;DR

This paper investigates the maximum number of paths of a fixed length in directed graphs with certain constraints, extends results to semirings, and poses an open problem on path algebra dimensions.

Contribution

It characterizes graphs that maximize path counts for fixed length and extends the results to semirings, also proposing an open problem on path algebra dimensions.

Findings

Identifies graphs maximizing path counts for fixed length.

Extends path counting results to semirings within non-negative reals.

Poses an open problem on the maximal dimension of path algebras.

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.