Triangle inequality for resistances

Given an electrical circuit each edge e of which is an isotropic conductor with a monomial conductivity function y∗ e = y r e/μ s e. In this formula, ye is the potential difference and y∗ e current in e, while μe is the resistance of e, while r and s are two strictly positive real parameters common for all edges. In 1987, Gvishiani and Gurvich [4] proved that, for every two nodes a, b of the circuit, the effective resistance μa,b is well-defined and for every three ordered nodes a, b, c the inequality μ s/r a,b ≤ μ s/r a,c + μ s/r c,b holds. It obviously implies the standard triangle inequality μa,b ≤ μa,c + μc,b when s ≥ r. In 1992, the same authors showed [5] that the equality holds if and only if c belongs to every path between a and b. Recently, Pavel Chebotarev has found several earlier works of 1967 by Gerald SubakSharpe [16, 17, 18] in which the inequality was shown for the case s = r = 1. Furthermore, it was rediscovered in 1993 by Douglas J. Klein and Milan Randić. In this report we provide the story of the considered inequality with more details.