  The total graph of $G$, $\mathcal T(G)$ is the graph whose set of vertices is
the union of the sets of vertices and edges of $G$, where two vertices are
adjacent if and only if they stand for either incident or adjacent elements in
$G$. Let $\mathcal{T}^1(G)=\mathcal{T}(G)$, the total graph of $G$. For
$k\geq2$, the $k\text{-}th$ iterated total graph of $G$, $\mathcal{T}^k(G)$, is
defined recursively as $\mathcal{T}^k(G)=\mathcal{T}(\mathcal{T}^{k-1}(G)).$ If
$G$ is a connected graph its diameter is the maximum distance between any pair
of vertices in $G$. The incidence energy $IE(G)$ of $G$ is the sum of the
singular values of the incidence matrix of $G$. In this paper for a given
integer $k$ we establish a necessary and sufficient condition under which
$diam(\mathcal{T}^{r+1}(G))>k-r,$ $r\geq0$. In addition, bounds for the
incidence energy of the iterated graph $\mathcal{T}^{r+1}(G)$ are obtained,
provided $G$ to be a regular graph. Finally, new families of non-isomorphic
cospectral graphs are exhibited.
