Expansion vs Sparsest cut
I think there may be multiple instances of such graphs. One pathological case that can be constructed is as follows 

Consider three disjoint vertex sets S,T,O. S is a single vertex of degree d+1 with all of edges going into the set O, i.e. there are no edges between S and T. Let O be a very large set and let the edges inside O be such that O is a clique. Now the idea is that we will set edges such that T has smaller expansion than S but T will have greater sparsity. So essentially such that S is the smallest sparsity set and T is the smallest expansion. Set up edges such that T is a clique and every vertex in T has d edges going into O. Suppose that |O| and |T| >> d

Now clearly T is the set with smallest expansion (of expansion d) (this is not hard to see because subsets of T and O have a lot of edges going out). Once again the smallest sparsity sets would be either T or S. Sparsity of S = d+1 / |O| + |T| and Sparsity of T = d / |O| + 1 . Given our chosen set of parameters. Sparsity of T > Sparsity of S