The University of Michigan Combinatorics Seminar


Abstract 

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 BC conjecture fails.
