A boolean circuit $f(x_1,\ldots,x_n)$ \emph{represents} a graph $G$ on $n$ vertices if for every input vector $a\in\{0,1\}^n$ with precisely two $1$'s in, say, positions $i$ and $j$, $f(a)=1$ precisely when $i$ and $j$ are adjacent in $G$; on inputs with more or less than two $1$'s the circuit can output ... more >>>

We consider the minimal number of AND and OR gates in monotone circuits for quadratic boolean functions, i.e. disjunctions of length-$2$ monomials. The single level conjecture claims that monotone single level circuits, i.e. circuits which have only one level of AND gates, for quadratic functions are not much larger than ... more >>>