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 arbitrary values.

We consider several types of boolean circuits (depth-$3$ circuits and

boolean formulas) and show that some explicit graphs cannot be

represented by small circuits. As a consequence we obtain that

an explicit boolean function in $2m$ variables cannot be computed

as an OR of fewer than $2^{\Omega(m)}$ products of linear forms

over $GF(2)$. Lower bounds for this model obtainable by previously

known (algebraic) arguments do not exceed $2^{O(\sqrt{m})}$.

We conclude with a graph-theoretic problem whose solution would have

some intriguing consequences in computational complexity.

Journal version published in: Combinatorics, Probability & Computing 15 (2006) 855-876.