Bridged graphs, circuits and Fibonacci numbers

Abstract

Series-parallel two-terminal graphs and corresponding unit resistor circuits are well explored. Here we expand the ideas to exclusive-bridged graphs and unit resistor circuits. We proof that both types of circuits have rational resistances whereof numerators and denominators of reduced fractions are smaller or equal to the Fibonacci number Fn+1. Series-parallel circuits satisfy another inequality. The sum of their numerator and denominator is smaller or equal to Fn+2. This is not true for exclusive-bridged circuits. The consequence is that combinations of these circuits or double-bridged circuits do not satisfy these inequalities. In a second part, counting series-parallel graphs and circuits is expanded to exclusive bridged graphs and circuits. This leads to new terms in two OEIS sequences.