  Graph eigenvalues play a fundamental role in controlling structural
properties, such as bisection bandwidth, diameter, and fault tolerance, which
are critical considerations in the design of supercomputing interconnection
networks. This motivates considering graphs with optimal spectral expansion,
called Ramanujan graphs, as potential candidates for interconnection networks.
In this work, we explore this possibility by comparing Ramanujan graph
properties against those of a wide swath of current and proposed supercomputing
topologies. We derive analytic expressions for the spectral gap, bisection
bandwidth, and diameter of these topologies, some of which were previously
unknown. We find the spectral gap of existing topologies are well-separated
from the optimal achievable by Ramanujan topologies, suggesting the potential
utility of adopting Ramanujan graphs as interconnection networks.
