In this talk, I will present the recent results regarding finding the optimal upper bound for the number of paths amongst acyclic, connected graphs with N edges. I will sketch the procedure of finding the graph that realizes this bound and show how to adapt these methods to find an optimal bound for Leavitt path algebras of a finite, acyclic, connected graph with N edges. The talk is based on a joint work with P. M. Hajac, M. Lowiel and E. A. Pacheco.