On a conjecture of Godsil concerning controllable random graphs

TL;DR

This paper proves Godsil's conjecture that almost all large simple graphs are controllable and extends the result to a broad class of Wigner random matrices, using advanced probabilistic methods.

Contribution

It confirms Godsil's conjecture and generalizes controllability results to Wigner matrices with new probabilistic techniques.

Findings

Almost all large simple graphs are controllable.

Controllability holds for a wide class of Wigner matrices.

Uses advanced Littlewood-Offord theory for proofs.

Abstract

It is conjectured by Godsil that the relative number of controllable graphs compared to the total number of simple graphs on n vertices approaches one as n tends to infinity. We prove that this conjecture is true. More generally, our methods show that the linear system formed from the pair (W, b) is controllable for a large class of Wigner random matrices W and deterministic vectors b. The proof relies on recent advances in Littlewood-Offord theory developed by Rudelson and Vershynin.