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We investigate a degree preserving variant of the $\Delta$-Y transformationwhich replaces a triangle with a new 6-valent vertex which has double edges tothe vertices that had been in the triangle. This operation is relevant forunderstanding scalar Feynman integrals in 6 dimensions. We study the structureof equivalence classes under this operation and its inverse, with particularattention to when the equivalence classes are finite, when they contain simple6-regular graphs, and when they contain doubled 3-regular graphs. The last ofthese, in particular, is relevant for the Feynman integral calculations and wemake some observations linking the structure of these classes to the Feynmanperiods. Furthermore, we investigate properties of minimal graphs in theseequivalence classes.

A degree preserving delta wye transformation with applications to 6-regular graphs and Feynman periods