The second eigenvalue of regular graphs of given girth
Journal of Combinatorial Theory Series B
Excluded minors, network decomposition, and multicommodity flow
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Small distortion and volume preserving embeddings for planar and Euclidean metrics
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Least-distortion Euclidean embeddings of graphs: products of cycles and expanders
Journal of Combinatorial Theory Series B
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(MATH) In this paper we partially prove a conjecture that was raised by Linial, London and Rabinovich in \cite{llr}. Let $G$ be a $k$-regular graph, $k \ge 3$, with girth $g$. We show that every embedding $f : G \to \ell_2$ has distortion $\Omega (\sqrt{g})$. The original conjecture which remains open is that the Euclidean distortion is bounded below by $\Omega(g)$. Two proofs are given, one based on semi-definite programming, and the other on Markov Type, a concept that considers random walks on metrics.