In recent years, graph theoretic considerations have become increasingly important in the design of HPC interconnection topologies. One approach is to seek optimal or near-optimal families of graphs with respect to a particular graph theoretic property, such as diameter. In this work, we consider topologies which optimize the spectral gap. In particular, we study a novel HPC topology, SpectralFly, designed around the Ramanujan graph construction of Lubotzky, Phillips, and Sarnak (LPS). We show combinatorial properties, such as diameter, bisection bandwidth, average path length, and resilience to link failure, of SpectralFly topologies are better than, or comparable to, similarly constrained DragonFly, SlimFly, and BundleFly topologies. Additionally, we simulate the performance of SpectralFly topologies on a representative sample of physics-inspired HPC workloads using the Structure Simulation Toolkit Macroscale Element Library simulator and demonstrate considerable benefit to using the LPS construction as the basis of the SpectralFly topology.