On the Banach space valued Azuma inequality and small set isoperimetry of Alon-Roichman graphs

TL;DR

This paper explores the relationship between graph expansion properties, Schatten norms, and Banach space inequalities, leading to improved bounds on small set isoperimetry in Alon-Roichman graphs using advanced probabilistic tools.

Contribution

It introduces a novel connection between Banach space valued Azuma inequalities and graph expansion, providing new bounds for small set isoperimetry in random Cayley graphs.

Findings

Improved bounds on small set isoperimetry of Abelian Alon-Roichman graphs.

Establishment of a link between Schatten norms and graph expansion.

Application of a variant of Azuma inequality in Banach spaces.

Abstract

We discuss the connection between the expansion of small sets in graphs, and the Schatten norms of their adjacency matrix. In conjunction with a variant of the Azuma inequality for uniformly smooth normed spaces, we deduce improved bounds on the small set isoperimetry of Abelian Alon-Roichman random Cayley graphs.