Graph equation

In graph theory, Graph equations are equations in which the unknowns are graphs. One of the central questions of graph theory concerns the notion of isomorphism. We ask: When are two graphs the same? (i.e., graph isomorphism) The graphs in question may be expressed differently in terms of graph equations.^[1]

What are the graphs (solutions) G and H such that the line graph of G is same as the total graph of H? (What are G and H such that L(G) = T(H) ?).

For example, G = K₃, and H = K₂ are the solutions of the graph equation L(K₃) = T(K₂) and G = K₄, and H = K₃ are the solutions of the graph equation L(K₄) = T(K₃).

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  :::${\displaystyle K_{2}}$
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  :::${\displaystyle K_{3}}$
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  :::${\displaystyle K_{4}}$

Note that T(K₃) is a 4-regular graph on 6 vertices.