  Let $G$ be a connected graph and let $\mathbb{X}$ be the set of projective
points defined by the column vectors of the incidence matrix of $G$ over a
field $K$ of any characteristic. We determine the generalized Hamming weights
of the Reed--Muller-type code over the set $\mathbb{X}$ in terms of graph
theoretic invariants. As an application to coding theory we show that if $G$ is
non-bipartite and $K$ is a finite field of ${\rm char}(K)\neq 2$, then the
$r$-th generalized Hamming weight of the linear code generated by the rows of
the incidence matrix of $G$ is the $r$-th weak edge biparticity of $G$. If
${\rm char}(K)=2$ or $G$ is bipartite, we prove that the $r$-th generalized
Hamming weight of that code is the $r$-th edge connectivity of $G$.
