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Abstract We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on n vertices with minimal degree linear in n is typically of order $\sqrt{n}$ . A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.more » « less
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Abstract We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb {R}^d$ generate high-rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets.more » « less
