Recurrence relations in m × 3 scaffolding and globe resistor networks

Abstract

Resistor networks are popular because they offer solvable models of transport between connected discrete points and can represent natural or artificial systems such as mycorhizzal networks or carbon composite chains. When the connectivity pattern is repeated, two-point resistances can be expressed by recurrence relations. Here, we illustrate this approach in the case of three-dimensional m × 3 scaffolding and globe networks, characterized by a repeated pattern along a three-fold invariant axis. We show that a first set of recurrence relations follows from three-fold invariance and Kennelly’s Y-Δ transform, providing the two-point resistance between any pair of neighbouring nodes, including the case of infinite networks. Using van Steenwijk’s method, a second set of recurrence relations is obtained between non-neighbouring nodes. Numerous explicit expressions are thus derived using elementary methods, which can be compared with the results of numerical codes or novel integration methods such as Tan’s recursion-transform. Having at hand general properties of these networks is useful to evaluate their capacity of representing natural or industrial systems.