{"paper":{"title":"$\\ell^p$-distortion and $p$-spectral gap of finite regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Alain Valette, Pierre-Nicolas Jolissaint","submitted_at":"2011-10-05T07:07:40Z","abstract_excerpt":"We give a lower bound for the $\\ell^p$-distortion $c_p(X)$ of finite graphs $X$, depending on the first eigenvalue $\\lambda_1^{(p)}(X)$ of the $p$-Laplacian and the maximal displacement of permutations of vertices. For a $k$-regular vertex-transitive graph it takes the form $c_p(X)^{p}\\geq diam(X)^{p}\\lambda_{1}^{(p)}(X)/2^{p-1}k$. This bound is optimal for expander families and, for $p=2$, it gives the exact value for cycles and hypercubes. As a new application we give a non-trivial lower bound for the $\\ell^2$-distortion of a family of Cayley graphs of $SL_n(q)$ ($q$ fixed, $n\\geq 2$) with r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0909","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}