TITLE: Characterization of non-isomorphic graphs but isomorphic total graphs?
QUESTION [5 upvotes]: 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.

REPLY [8 votes]: It is not difficult to see that both $G$ and $L(G)$ are disjoint induced subgraphs of the graph $T(G)$. For your question, in the below paper:
[1] M. Behzad and H. Radjavi, The total group of a graph, Proc. Amer. Math. Soc.
19 (1968), 158-163. MR 36 #1358.
It is proved that, the graph $G$ is isomorphic to the graph $H$ if and only if $T(G)$ is isomorphic to $T(H)$.