  The line graph $\Gamma$ of a multi-graph $\Delta$ is the graph whose vertices
are the edges of $\Delta$, where two such edges are adjacent if and only if
they meet in a single vertex of $\Delta$.
  We provide several characterizations of such line graphs and in particular
show that a graph is a line graph if and only if it does not contain one of
$33$ graphs, all of which correspond to bases of anisotropic vectors of a
$6$-dimensional orthogonal geometry of $-$-type over a field with two elements,
or, equivalently, to sets of $6$ generating reflections in the Weyl group of
type $E_6$.
