Fourier analysis on finite groups and the Lov\'asz theta-number of Cayley graphs

TL;DR

This paper uses Fourier analysis on finite groups to simplify the computation of the Lovász theta-number for Cayley graphs, applying it to verify cases of a conjecture related to intersecting families of permutations.

Contribution

It introduces simplified formulas for the Lovász theta-number of Cayley graphs and applies them to verify cases of a conjecture on intersecting permutation families, including a new q-analog.

Findings

Simplified formulas for Lovász theta-number of Cayley graphs

Verification of cases of a conjecture on intersecting permutation families

Introduction of a q-analog of k-intersecting families

Abstract

We apply Fourier analysis on finite groups to obtain simplified formulations for the Lov\'asz theta-number of a Cayley graph. We put these formulations to use by checking a few cases of a conjecture of Ellis, Friedgut, and Pilpel made in a recent article proving a version of the Erd\H{o}s-Ko-Rado theorem for $k$-intersecting families of permutations. We also introduce a $q$-analog of the notion of $k$-intersecting families of permutations, and we verify a few cases of the corresponding Erd\H{o}s-Ko-Rado assertion by computer.