The maximal dimensions of path and graph algebras

We consider the class of acyclic connected directed graphs with $N\ge 1$. In this paper we find the optimal upper bound for the number of paths amongst acyclic, connected graphs with $N$ edges. We prove that it is in fact optimal by finding an acyclic, connected graph with $N$ edges that realizes this bound. We then adapt these methods to find an optimal bound for Leavitt path algebras over a finite, acyclic, connected graph with $N$ edges.