Matrix analysis
The Johnson-Lindenstrauss Lemma and the sphericity of some graphs
Journal of Combinatorial Theory Series A
Geometrical embeddings of graphs
Discrete Mathematics
Dispersed points and geometric embedding of complete bipartite graphs
Discrete & Computational Geometry
On embedding of graphs into Euclidean spaces of small dimension
Journal of Combinatorial Theory Series B
Applications of cut polyhedra—II
Journal of Computational and Applied Mathematics
Unit disk graph recognition is NP-hard
Computational Geometry: Theory and Applications - Special issue on geometric representations of graphs
Finite metric spaces: combinatorics, geometry and algorithms
Proceedings of the eighteenth annual symposium on Computational geometry
Introduction to Coding Theory
Lectures on Discrete Geometry
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Betweenness parameterized above tight lower bound
Journal of Computer and System Sciences
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We consider monotone embeddings of a finite metric space into low-dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l2n, while (in a sense to be made precise later), for almost every n-point metric space, every monotone map must be into a space of dimension Ω(n) (Lemma 3).It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on sphericity of graphs, which suggest one example of such a metric space--that is defined by a complete bipartite graph. We prove that an δn-regular graph of order n, with bounded diameter has sphericity Ω(n/(λ2+1)), where λ2 is the second largest eigenvalue of the adjacency matrix of the graph, and 02 must be constant. We show that if the second eigenvalue of an n/2-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only o(n2) entries (Theorem 5). Furthermore, for any 02, there are only finitely many δn-regular graphs with second eigenvalue at most λ2 (Corollary 4).