$\ell^p$-distortion and $p$-spectral gap of finite regular graphs

TL;DR

This paper establishes a lower bound for the $ ext{l}^p$-distortion of finite graphs based on the $p$-spectral gap and graph diameter, with applications to expander graphs, cycles, hypercubes, and Cayley graphs of $SL_n(q)$.

Contribution

It introduces a new lower bound for $ ext{l}^p$-distortion involving the $p$-spectral gap and vertex displacement, providing exact results for specific graph families.

Findings

Bound is optimal for expander families.

Exact $ ext{l}^2$-distortion for cycles and hypercubes.

New lower bound for Cayley graphs of $SL_n(q)$.

Abstract

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 respect to a standard two-element generating set.