On the log concavity of reliability and matroidal sequences (Q1325759)

The reliability of a graph \(G\) is the probability that \(G\) is connected, given that the edges are independently operational with probability \(p\). This is known to be a polynomial in \(p\), and various sequences associated with this polynomial have been conjectured to be unimodal, and indeed, log concave. It is shown that for any graph \(G\), there is a subdivision for which the log concavity conjectures all hold. Further, evidence is provided for the well-known conjecture of the log concavity of the independent set numbers of a matroid.