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 $G$. 



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> Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)$?
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Cross-posted at [MSE](https://math.stackexchange.com/questions/1633038/characterization-of-non-isomorphic-graphs-but-isomorphic-total-graphs).



Characterization of non-isomorphic graphs but isomorphic total graphs?
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 $T(H)$.


