Representation Theoretic Techniques for Independence Bounds of Cayley Graphs

The independence number of Cayley graphs has proved to be an important topic in discrete mathematics, information theory, and computer science [18, 2]. In early 2017, Kane, Lovett, and Rao provided a novel technique to compute such bounds via representation theory. While earlier works had used the representation theory of abelian groups to analyze independent sets [29, 3, 14], Kane et al. were the first to bring analysis of a non-abelian group, Sn, to the table, proving an independence bound for the Birkhoff graph. This work aims to provide background, intuition, and novel applications of Kane et al’s new technique in order to show its efficacy and robustness across a variety of circumstances. In particular, we first show how the KLR technique builds off of the Hoffman bound which recovers special cases of closed-form bounds given by Delsarte’s linear program on abelian Cayley graphs associated to Aq (n, d). Second, we explain how the KLR technique employs structure vs randomness for non-abelian groups, and show that the technique generalizes to the hyperoctahedral group.