Abstract
The reliability of a graph G is the probability that G is connected, given that edges are independently operational with probability p. This is known to be a polynomial in p, and the location of the roots of these functions is discussed. In particular, it is conjectured that the roots of the reliability polynomial of any connected graph lie in the disc $| z - 1 | \leq 1$, and evidence for this conjecture is provided. It is shown that all real roots lie in $\{ 0 \} \cup ( 1,2 ]$ and that every graph has a subdivision for which the roots of the reliability polynomial lie in the conjectured disc.
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