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 transforms**, 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 a loop.

The Y-Δ transform is a special case of the star-mesh transform in which the center node is not a boundary node and has the degree *3*.