Characterization of bipartite Cayley graphs in terms of Boolean functions

Abstract

A Boolean function on n-variables is a function f from Vn into field GF(2) and we denote the set of all n-variable Boolean functions by Fn. For every f ∈Fn, the Walsh transform of f is defined as Wf (a) = Σx∈GF(2n)f(x)(−1)a.x. For the Boolean function f, the list [Wf (0),…, (2n −1)] is called the Walsh spectrum of f. The aim of this paper is to study the Walsh spectrum of Cayley graphs in terms of Boolean functions. We also characterize some Boolean functions whose related bipartite Cayley graphs have four distinct eigenvalues.