Metric and ultrametric spaces of resistances

Consider an electrical circuit, each edge ee of which is an isotropic conductor with a monomial conductivity function ye∗=yer/μes. In this formula, yeye is the potential difference and ye∗ current in ee, while μeμe is the resistance of ee; furthermore, rr and ss are two strictly positive real parameters common for all edges. In particular, the case r=s=1r=s=1 corresponds to the standard Ohm’s law.In 1987, Gvishiani and Gurvich [A.D. Gvishiani, V.A. Gurvich, Metric and ultrametric spaces of resistances, in: Communications of the Moscow Mathematical Society, Russian Math. Surveys 42 (6 (258)) (1987) 235–236] proved that, for every two nodes a,ba,b of the circuit, the effective resistance μa,bμa,b is well-defined and for every three nodes a,b,ca,b,c the inequality μa,bs/r≤μa,cs/r+μc,bs/r holds. It obviously implies the standard triangle inequality μa,b≤μa,c+μc,bμa,b≤μa,c+μc,b whenever s≥rs≥r. For the case s=r=1s=r=1, these results were rediscovered in the 1990s. Now, after 23 years, I venture to reproduce the proof of the original result for the following reasons:
• It is more general than just the case r=s=1r=s=1 and one can get several interesting metric and ultrametric spaces playing with parameters rr and ss. In particular, (i) the effective Ohm resistance, (ii) the length of a shortest path, (iii) the inverse width of a bottleneck path, and (iv) the inverse capacity (maximum flow per unit time) between any pair of terminals aa and bb provide four examples of the resistance distances μa,bμa,b that can be obtained from the above model by the following limit transitions: (i) r(t)=s(t)≡1r(t)=s(t)≡1, (ii) r(t)=s(t)→∞r(t)=s(t)→∞, (iii) r(t)≡1,s(t)→∞r(t)≡1,s(t)→∞, and (iv) r(t)→0,s(t)≡1r(t)→0,s(t)≡1, as t→∞t→∞. In all four cases the limits μa,b=limt→∞μa,b(t)μa,b=limt→∞μa,b(t) exist for all pairs a,ba,b and the metric inequality μa,b≤μa,c+μc,bμa,b≤μa,c+μc,b holds for all triplets a,b,ca,b,c, since s(t)≥r(t)s(t)≥r(t) for any sufficiently large tt. Moreover, the stronger ultrametric inequality μa,b≤max(μa,c,μc,b)μa,b≤max(μa,c,μc,b) holds for all triplets a,b,ca,b,c in examples (iii) and (iv), since in these two cases s(t)/r(t)→∞s(t)/r(t)→∞, as t→∞t→∞.
• Communications of the Moscow Math. Soc. in Russ. Math. Surveys were (and still are) strictly limited to two pages; the present paper is much more detailed.Although a translation in English of the Russ. Math. Surveys is available, it is not free in the web and not that easy to find.
• The last but not least: priority.