Almost all graphs of order n have no cospectral mates with height o((n / ln n)^{1/10}).
[VVW26] Nils Van de Berg and Alexander Van Werde
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For n x n i.i.d. Bernoulli(p) matrices, P(corank >= k) = (1-p + o_n(1))^{k n} when k = O(sqrt(log n)).
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Almost all graphs have no cospectral mates with height relative small to its order
Almost all graphs of order n have no cospectral mates with height o((n / ln n)^{1/10}).