A lower bound on the average physical length of edges in the physical realization of graphs

The stereo-realization of a graph is the assignment of positions in Cartesian space to each of its vertices such that vertex density is bounded. A bound is derived on the average edge length in such a realization. It is similar to an earlier reported result, however the new bound can be applied to graphs for which the earlier result is not well suited. A more precise realization definition is also presented. The bound is applied to d-dimensional realizations of de Bruijn graphs, yielding an edge length of Ω((1 - 2-d)rn/d/(2n)), where r is the radix (number of distinct symbols) and n is the number of graph dimensions (number of symbol positions). The bound is also applied to shuffle-exchange graphs; for such graphs with small radix the edge-length bound is 2/3lσ + 1/3lε (1 - 2-d)rn/d/(2(n(2 - 1/r) - 1)), where r is the radix, n is the number of graph dimensions, lσ is the average length of shuffle edges, and lε is the average length of exchange edges. © World Scientific Publishing Company