The number of realisations of a random graph

Abstract

Determining the number of realisations, up to isometries, of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that, asymptotically almost surely, the $d$-dimensional complex realisation number for each $n$-vertex graph in an Erdős-Rényi random graph process is either infinite or equal to $2^{n-t}$ where $t$ is the size of the $(d+1)$-core; moreover this number coincides exactly with the real realisation number for such graphs. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ positive semi-definite matrix completion problem with randomly selected non-diagonal unknown entries.

20 pages, 2 figures. Improved presentation of results and fixed minor errors in Appendix