Networks, Polynomials, and Matroids

Consider a connected undirected (multi)graph G with a complex admittance (inverse resistance) y_e associated to each edge. For any two vertices a and b of G, the effective admittance of (G,{y_e}) between a and b is a rational function of the y_e with the following property: if all the y_e have positive real part, then the value of the function has positive real part as well. These are classical results in the electrical engineering literature. In this talk I will report on the current state of ongoing joint research with Y.B. Choe, J.G. Oxley, and A.D. Sokal, in which we generalize these ideas from graphs to more general matroids. We do have some necessary and some sufficient conditions, but the relevant classes of matroids seem to be very difficult to characterize. There is also a connection with the 1992 conjecture of Brown and Colbourn on location of zeros of reliability polynomials -- in particular, we find examples of matroids for which the strong form of the B-C conjecture fails.