This is a drawing of Stewart’s cube. At each corner, the sum of the weights of the incoming edges is 83. Each edge has a prime weight, and all the weights are distinct.
I wondered if Stewart’s cube achieved its structure in the cheapest way. I considered two objectives: (1) minimize the sum of the edges incident to a vertex, and (2) minimize the weight of the worst edge. In both cases, the following cube is optimal.
It improves the first objective from 83 to 77, and the second objective from 61 to 53. To solve the problems, I modeled them as integer programs, and used Gurobi as solver. I’m currently running code for higher-dimensional cubes.
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Gurobi had problems with higher-dimensional cubes. The LP relaxation is not very good, and my formulation had a lot of symmetry.