Characterization of non-isomorphic graphs but isomorphic total graphs?

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 $T(H)$?

• mathoverflow.net/questions/229696/… – sriram Jan 31 '16 at 18:20
• Since you got an answer on mathoverflow.net, you could mention this here in math.stackexchange.com ? – Moritz Jan 31 '16 at 18:29
• That is what I provided the link – sriram Feb 2 '16 at 7:04