  Let $G$ be a simple graph on the vertex set $V(G) = [n] = \{1,...,n\}$ and
edge ideal $E(G)$. We consider the class of closed graphs. A closed graph is a
simple graph satisfying the following property: for all edges $\{i, j\}$ and
$\{k, \ell\}$ with $i < j$ and $k < \ell$ one has $\{j, \ell\}\in E(G)$ if $i =
k$, and $\{i, k\}\in E(G)$ if $j = \ell$. We state some criteria for the
closedness of a graph $G$ that do not depend necessarily from the labelling of
its vertex set. Consequently, if $S = K[x_1,..., x_n, y_1,..., y_n]$ is a
polynomial ring in $2n$ variables with coefficients in a field $K$, we obtain
some criteria for the Koszulness of the quotient algebra $S /J_G$, where $J_G$
is the binomial edge ideal of $S$ i.e. the ideal generated by the binomials
$f_{ij} = x_iy_j - x_jy_i$ such that $i<j$ and $\{i,j\}$ is an edge of $G$
(\cite{HH}).
