### DIMACS - Graduate Student Combinatorics Seminar

Title: Metric and Ultrametric Spaces of Resistances

Speaker: Vladimir Gurvich, RUTCOR, Rutgers University

Date: Wednesday, February 9, 2011 12:10pm

Location: Graduate Student Lounge, 7th Floor, Hill Center, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Given an electrical circuit each edge e of which is an isotropic conductor with a monomial conductivity function ye* = yer / ”es, where, ye is the potential difference and ye* current in e, while ”e is the resistance of e; furthermore, r and s are two strictly positive real parameters common for all edges. In particular, r = s = 1 is the standard Ohm law; r = 1/2 is typical for hydraulics or gas dynamics.

For every two nodes a, b the effective resistance ”a,b is well-defined and for every three nodes a,b,c the inequality ”a,bs/r = ”a,cs/r + ”c,bs/r holds. It implies the standard metric inequality whenever s = r and ultramentric one when s(t)/r(t) ? 8, as t ? 8.

One can get several metric and ultrametric spaces playing with parameters r and s. In particular, the effective Ohm resistance, the length of a shortest path, the inverse width of a bottleneck path, and the inverse capacity (maximum flow per unit time) between any pair of terminals.