Vertex transitivity and distance metric of the quad-cube.

The quad-cube is a special case of the metacube that itself is derivable from the hypercube. It is amenable to an application as a network topology, especially when the node size exceeds several million. This paper presents the following welcome properties of the graph, relating to its structure: (1) vertex transitivity that facilitates the working of an algorithm meant for a "local" context in the global context as well, and (2) an exact formula for the distance metric, which leads to a precise result on the distance-wise vertex distribution of the graph and an exact formula for the average vertex distance. Remarkably, the vertex distribution of the quad-cube resembles, to a large extent, the vertex distribution of the twin copies of a hypercube. In a parallel study, the author recently reported similar results with respect to the dual-cube (Jha in J Supercomput 78:17758–17775, 2022) [ABSTRACT FROM AUTHOR]