Girth and euclidean distortion

(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.

[1]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[2]  P. Enflo.  On the nonexistence of uniform homeomorphisms betweenLp-spaces , 1970 .

[3]  J. Matousek,et al.  On embedding trees into uniformly convex Banach spaces , 1999 .

[4]  Nathan Linial,et al.  Least-Distortion Euclidean Embeddings of Graphs: Products of Cycles and Expanders , 2000, J. Comb. Theory, Ser. B.

[5]  J. Bourgain.  The metrical interpretation of superreflexivity in banach spaces , 1986 .

[6]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[7]  Jean Bourgain,et al.  On type of metric spaces , 1986 .

[8]  Philip N. Klein,et al.  Excluded minors, network decomposition, and multicommodity flow , 1993, STOC.

[9]  Notes on an inequality by pisier for functions on the discrete cube , 2000 .

[10]  Anupam Gupta,et al.  Cuts, Trees and ℓ1-Embeddings of Graphs* , 2004, Comb..

[11]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

[12]  M. Talagrand.  Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem , 1993 .

[13]  Yair Bartal,et al.  On approximating arbitrary metrices by tree metrics , 1998, STOC '98.

[14]  Patrick Solé,et al.  The second eigenvalue of regular graphs of given girth , 1992, J. Comb. Theory, Ser. B.

[15]  Keith Ball,et al.  Markov chains, Riesz transforms and Lipschitz maps , 1992 .

[16]  G. Pisier.  Probabilistic methods in the geometry of Banach spaces , 1986 .

[17]  Satish Rao,et al.  Small distortion and volume preserving embeddings for planar and Euclidean metrics , 1999, SCG '99.

[18]  J. Matousek,et al.  On embedding expanders into ℓp spaces , 1997 .