Sufficient conditions for bipartite rigidity, symmetric completability and hyperconnectivity of graphs

We consider three matroids defined by Kalai in 1985: the symmetric completion matroid $\mathcal{S}_d$ on the edge set of a looped complete graph; the hyperconnectivity matroid $\mathcal{H}_d$ on the edge set of a complete graph; and the birigidity matroid $\mathcal{B}_d$ on the edge set of a complete bipartite graph. These matroids arise in the study of low rank completion of partially filled symmetric, skew-symmetric and rectangular matrices, respectively. We give sufficient conditions for a graph $G$ to have maximum possible rank in these matroids. For $\mathcal{S}_d$ and $\mathcal{H}_d$, our conditions are in terms of the minimum degree of $G$ and are best possible. For $\mathcal{B}_d$, our condition is in terms of the connectivity of $G$. Our results have several implications for the unique completability of low-rank matrices. In particular, they imply that: almost all sufficiently large $n \times n$ positive semidefinite matrices of rank $d$ are uniquely determined by any subset of their entries which includes at least $(n + d + 1)/2$ entries from each row; almost all $m \times n$ matrices of rank $d$ are uniquely determined by any subset of their entries whose positions define a spanning subgraph of $K_{m,n}$ which is $k_d$-connected, for some constant $k_d=\mbox{O}(d^3)$.