A Combinatorial Proof Of Rayleigh Monotonicity For Graphs

We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs. Consider a (linear, resistive) electrical network – this is a connected graph G = (V,E) and a set of positive real numbers y = {ye : e ∈ E} indexed by E. In this paper we allow graphs to have loops and/or multiple edges. The value of ye is interpreted as the electrical conductance of a wire joining the vertices incident with e. For any edge e ∈ E, there is a simple formula for the effective conductance Ye(G;y) of the rest of the graph G r {e}, measured between the ends of e. This is due to Kirchhoff [11] and is also known as Maxwell’s Rule [12]. For a subset S ⊆ E, let