# Ten years of work on the Brown-Colbourn conjecture

## ABSTRACT

The reliability polynomial $R(G;q)$ of a (finite, connected, loopless, multi-) graph is the probability that if each edge is deleted independently with probability $q$, then the remaining spanning subgraph is still connected. In 1992, Brown and Colbourn conjectured that every complex zero of the polynomial $R(G;q)$ lies in the closed unit disc $|q|\leq 1$. In this talk I will review the progress which has been made towards understanding this conjecture over the past decade, including work of Brown, Colbourn, Sokal, Chari, Choe, Royle, and others. Possible generalizations to matroids, and implications for the $f$--vectors of matroids will also be discussed.

Speaker's Contact Info: dgwagner(at-sign)pythagoras.math.uwaterloo.ca