  We wish to bring attention to a natural but slightly hidden problem, posed by
Erd\H{o}s and Ne\v{s}et\v{r}il in the late 1980s, an edge version of the
degree--diameter problem. Our main result is that, for any graph of maximum
degree $\Delta$ with more than $1.5 \Delta^t$ edges, its line graph must have
diameter larger than $t$. In the case where the graph contains no cycle of
length $2t+1$, we can improve the bound on the number of edges to one that is
exact for $t\in\{1,2,3,4,6\}$. In the case $\Delta=3$ and $t=3$, we obtain an
exact bound. Our results also have implications for the related problem of
bounding the distance-$t$ chromatic index, $t>2$; in particular, for this we
obtain an upper bound of $1.941\Delta^t$ for graphs of large enough maximum
degree $\Delta$, markedly improving upon earlier bounds for this parameter.
