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We consider the difference between the two lowest eigenvalues (thefundamental gap) of a Schr\"{o}dinger operator acting on a class of graphs. Inparticular, we derive tight bounds for the gap of Schr\"{o}dinger operatorswith convex potentials acting on the path graph. Additionally, for thehypercube graph, we derive a tight bound for the gap of Schr\"{o}dingeroperators with convex potentials dependent only upon vertex Hamming weight. Ourproof makes use of tools from the literature of the fundamental gap theorem asproved in the continuum combined with techniques unique to the discrete case.We prove the tight bound for the hypercube graph as a corollary to our pathgraph results.

The Fundamental Gap for a Class of Schr\"{o}dinger Operators on Path and Hypercube Graphs