Miller, Teng, Thurston, and Vavasis proved a geometric separator theorem which implies that the k-nearest neighbor graph (k-NNG) of every set of n points in R d has a balanced vertex separator of size O(n 1−1/d k 1/d ). Spielman and Teng then proved that the Fiedler value -the second smallest eigenvalue of the Laplacian matrix -of the k-NNG of any n In this paper, we extend these two results to nearest neighbor graphs in a metric space with a finite doubling dimension and in a metric space that

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... is nearly-Euclidean. We prove that for every l > 0, if (X, dist) forms a metric space with doubling dimension γ , then the k-NNG of every set P of n points in X has a vertex separator of size O(k 2 l(64l + 8) 2γ log 2 L S log n + n l ), where L and S are, respectively, the maximum and minimum distances between any two points in P. We show how to use the singular value decomposition method to approximate a k-NNG in a nearly-Euclidean space by a Euclidean k-NNG. This approximation enables us to obtain an upper bound on the Fiedler value of k-NNGs in a nearly-Euclidean space.