## Tbilisi Mathematical Journal

- Tbilisi Math. J.
- Volume 5, Issue 2 (2012), 19-35.

### On the Walsh-Hadamard transform of monotone Boolean functions

Charles Celerier, David Joyner, Caroline Melles, and David Phillips

#### 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 $n>2$.

#### Article information

**Source**

Tbilisi Math. J., Volume 5, Issue 2 (2012), 19-35.

**Dates**

First available in Project Euclid: 12 June 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.tbilisi/1528768900

**Mathematical Reviews number (MathSciNet)**

MR3055512

**Zentralblatt MATH identifier**

1283.06021

**Subjects**

Primary: 06E30: Boolean functions [See also 94C10]

Secondary: 94C10: Switching theory, application of Boolean algebra; Boolean functions [See also 06E30]

**Keywords**

Boolean functions Walsh-Hadamard transforms monotone Boolean functions

#### Citation

Celerier, Charles; Joyner, David; Melles, Caroline; Phillips, David. On the Walsh-Hadamard transform of monotone Boolean functions. Tbilisi Math. J. 5 (2012), no. 2, 19--35. https://projecteuclid.org/euclid.tbilisi/1528768900