I'm confused about the realtionship between dual graphs and the so called star-mesh transformation: https://en.wikipedia.org/wiki/Star-mesh_transform.
Take a simple triangle, its dual graph looks like three lines meeting at a point (connected to another external point). The star-mesh duality tells us exactly that also, i.e. apply the transformation on a triangle and you'll get a 3 pointed star.
Now, if we take a a graph with 4 points that's maximally connected so that it has 6 edges (a tetrahedron), when we draw the dual graph it will look something like a butterfly (try it!). However according to the star-mesh duality, a 4 pointed star is what is equivalent to a tetrahedral a graph.
The star mesh duality for a triangle and tetrahedron
What exactly is going on here, do they just superficially seem related and can both things be true? The problem to me seems to be that for $N>3$ the star-mesh duality relates diagrams with different numbers of edges, whereas the dual diagram preserves the number of edges.
I have such a tetrahedral graph where the 6 edges are weighted, I wanted to use the star-mesh transform to find what the edges of the dual graph were (in terms of the original weights), but since this star-mesh duality doesn't seem to produce the dual graph I am not sure what one can use to do this.