The generic-rank conjecture for bipartite circulant graphs

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Let G(n,l)G(n,l) be a bipartite circulant graph of the form considered in the paper, and let the generic completion rank be the rank attained by a generic matrix completion. The dimension-count prediction is

n−⌊n2−nl⌋.n- \left\lfloor\sqrt{n^2-nl}\right\rfloor.

Bipartite circulant generic-rank conjecture. Every graph G(n,l)G(n,l) has generic completion rank predicted by the dimension count; equivalently,

gcr⁡(G(n,l))=n−⌊n2−nl⌋.\operatorname{gcr}(G(n,l))=n-\left\lfloor\sqrt{n^2-nl}\right\rfloor.

The paper proves the dimension-count prediction for a subset of these graphs and conjectures it for all bipartite circulant graphs of this form. The claim is presented as unresolved.

References

Primary source

Daniel Irving Bernstein, Grigoriy Blekherman and Rainer Sinn, “Typical and Generic Ranks in Matrix Completion”, arXiv:1802.09513 (2019).

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