  We study the space complexity of sketching cuts and Laplacian quadratic forms
of graphs. We show that any data structure which approximately stores the sizes
of all cuts in an undirected graph on $n$ vertices up to a $1+\epsilon$ error
must use $\Omega(n\log n/\epsilon^2)$ bits of space in the worst case,
improving the $\Omega(n/\epsilon^2)$ bound of Andoni et al. and matching the
best known upper bound achieved by spectral sparsifiers. Our proof is based on
a rigidity phenomenon for cut (and spectral) approximation which may be of
independent interest: any two $d-$regular graphs which approximate each other's
cuts significantly better than a random graph approximates the complete graph
must overlap in a constant fraction of their edges.
