The effective conductivities b/w boundary nodes and the Dirichlet-to-Neumann operator of a network are invariant under the following star-mesh and Y-Δ transform, illustrated by the following drawings from Wikipedia:
Exercise (**). Let d be a diagonal entry of the Kirchhoff matrix K of a network G, corresponding to an interior node. Use the Schur complement formula
for the Dirichlet-to-Neumann operator to prove the invariance.
The rules for replacing conductors in series or parallel connection by a single electrically equivalent conductor follow from the invariance property of the Y-Δ transform and can be viewed as its special cases, as also erasing an edge w/an end point of degree 1 and erasing an edge, which endpoints coincide.
The Y-Δ transform is a special case of the star-mesh transform in which the center node has the degree 3.