Given a graph $G$, the *total graph of $G$*, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$.  

>Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)$?

Cross-posted at [MSE](https://math.stackexchange.com/questions/1633038/characterization-of-non-isomorphic-graphs-but-isomorphic-total-graphs).