In this paper, we develop the groundwork for a graph theoretic toy model of supersymmetric quantum mechanics. Using discrete Witten-Morse theory, we demonstrate that finite graphs have a natural supersymmetric structure and use this structure to incorporate supersymmetry into an existing model of graph quantum mechanics. We prove that although key characteristics of continuum supersymmetric systems are preserved on finite unweighted graphs, supersymmetry cannot be spontaneously broken. Finally, we prove new results about the behavior of supersymmetric graph quantum systems under edge rewiring.