  In this paper we explore some results concerning the spread of the line and
the total graph of a given graph. In particular, it is proved that for an
$(n,m)$ connected graph $G$ with $m > n \geq 4$ the spread of $G$ is less than
or equal to the spread of its line graph, where the equality holds if and only
if $G$ is regular bipartite. A sufficient condition for the spread of the graph
not be greater than the signless Laplacian spread for a class of non bipartite
and non regular graphs is proved. Additionally, we derive an upper bound for
the spread of the line graph of graphs on $n$ vertices having a vertex (edge)
connectivity less than or equal to a positive integer $k$. Combining techniques
of interlacing of eigenvalues, we derive lower bounds for the Laplacian and
signless Laplacian spread of the total graph of a connected graph. Moreover,
for a regular graph, an upper and lower bound for the spread of its total graph
is given.
