Let G be a k-regular graph, k ≥ 3, with girth g. We prove that every embedding f : G → l2 has distortion Ω(√g). Two proofs are given, one based on Markov type [B] and the other on quadratic programming. In the core of both proofs are some Poincaré-type inequalities on graph metrics.
Bibliographical noteFunding Information:
N.L. was supported in part by grants from the Israel Science Foundation and the Binational Science Foundation Israel-USA. A.N. was supported in part by the Binational Science Foundation Israel-USA and the Clore Foundation. This work is part of a Ph.D. thesis being prepared under the supervision of Professor Joram Lindenstrauss.