Abstract: The Perron-Frobenius Theorem is a powerful result in linear algebra stating that a real, square matrix with strictly positive entries has a unique largest eigenvalue, and its corresponding eigenvector can be chosen such that it is composed of positive entries. The theorem has several variants: notably, an analogous statement for nonnegative matrices. Moreover, it has important consequences in the analysis of iterative methods in numerical analysis, and it possesses a large number of interesting applications to dynamical systems, probability theory, social networks, and combinatorics. Specifically in this talk, we will formally present the theorem and then explore its application to path counting problems in graph theory.