On the Walsh-Hadamard transform of monotone Boolean functions

Abstract

Let $f:GF(2)^n \rightarrow GF(2)$ be a monotone Boolean function. Associated to $f$ is the Cayley graph $X$ whose vertices correspond to points of $GF(2)^n$ and whose edges correspond to pairs of vectors $(v,w)$ whose sum is in the support of $f$. The spectrum of $X$ (the set of eigenvalues of its adjacency matrix) can be computed in terms of the Walsh-Hadamard transform of $f$. We show that if $f$ is atomic, the adjacency matrix of $X$ is singular if and only if the support of $f$ has an even number of elements. We ask whether it is true that for every even monotone function the adjacency matrix of the Cayley graph must be singular. We give an example in dimension $n=6$ to show that the answer to this question is no. We use Sage to compute some examples of monotone Boolean functions, their Cayley graphs, and the graph spectra. We include some interesting characterizations of monotone functions. We give some conditions on a monotone function that imply that the function is not bent. Finally, we ask whether it is true that no even monotone function is bent, for $n>2$.