Generalized bent Boolean functions and strongly regular Cayley graphs

TL;DR

This paper explores the relationship between generalized bent Boolean functions and strongly regular Cayley graphs, providing new characterizations and properties linking spectral flatness to graph regularity.

Contribution

It introduces a new notion of strong regularity for edge-weighted Cayley graphs associated with generalized Boolean functions and characterizes quartic gbent functions via this property.

Findings

Connected strong regularity with flat Walsh-Hadamard spectrum.

Characterized quartic gbent functions through Cayley graph properties.

Established properties of the associated Cayley graphs.

Abstract

In this paper we define the (edge-weighted) Cayley graph associated to a generalized Boolean function, introduce a notion of strong regularity and give several of its properties. We show some connections between this concept and generalized bent functions (gbent), that is, functions with flat Walsh-Hadamard spectrum. In particular, we find a complete characterization of quartic gbent functions in terms of the strong regularity of their associated Cayley graph.