Monotone maps, sphericity and bounded second eigenvalue

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.T o 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 0 < δ ≤ 1/2 is constant (Theorem 4). We also show that while random graphs have linear sphericity, there are quasi-random graphs of logarithmic sphericity (Lemma 7). For the above bound to be linear, λ₂ 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 0 < δ < 1/2, and λ₂, there are only finitely many δn-regular graphs with second eigenvalue at most λ₂ (Corollary 4).