Metric and ultrametric inequalities for resistances in directed graphs

Abstract

Consider an electrical circuit $G$ each directed edge $e$ of which is a semiconductor with a monomial conductance function $y_e^* = f_e(y_e) = y_e^s / μ_e^r$ if $y_e \geq 0$ and $y_e^* = 0$ if $y_e \leq 0$. Here $e$ is a directed edge, $y_e$ is the potential difference (voltage), $y_e^*$ is the current in $e$, and $μ_e$ is the resistance of $e$; furthermore, $r$ and $s$ are two strictly positive real parameters common for all edges. In particular, case $r = s = 1$ corresponds to the Ohm law, while $r = \frac{1}{2}, s =1$ may be interpreted as the square law of resistance typical for hydraulics and gas dynamics. We will show that for every ordered pair of nodes $a, b$ of the circuit, the effective resistance $μ_{a,b}$ is well-defined. In other words, any two-pole network with poles $a$ and $b$ can be effectively replaced by two oppositely directed edges, from $a$ to $b$ of resistance $μ_{a,b}$ and from $b$ to $a$ of resistance $μ_{b,a}$. Furthermore, for every three nodes $a, b, c$ the inequality $μ_{a,c}^{s/r} + μ_{c,b}^{s/r} \geq μ_{a,b}^{s/r}$ holds, in which the equality is achieved if and only if every directed path from $a$ to $b$ contains $c$. MSC classes: 11J83, 90C25, 94C15,94C99