We study Nordhaus–Gaddum problems for Kemeny’s constant K ( G ) of a connected graph G . We prove bounds on min { K ( G ) , K ( G ¯ ) } and the product K ( G ) K ( G ¯ ) for various families of graphs. In particular, we show that if the maximum degree of a graph G on n vertices is n − O ( 1 ) or n − Ω ( n ) , then min { K ( G ) , K ( G ¯ ) } is at most O ( n ) .

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. 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 on a representative sample of micro-benchmarks using the Structure Simulation Toolkit Macroscale Element Library simulator and study cost-minimizing layouts, demonstrating considerable benefit of the SpectralFly topology.