Regular Article: On the Log Concavity of Reliability and Matroidal Sequences

Authors:
J. I. Brown;C. J. Colbourn
Affiliations:
York Univ, Dept Math & Stat, N York M3J 1P3, ON, Canada and Univ Waterloo, Dept Combinator & Optimizat, Waterloo N2L 3G1, ON, Canada;York Univ, Dept Math & Stat, N York M3J 1P3, ON, Canada and Univ Waterloo, Dept Combinator & Optimizat, Waterloo N2L 3G1, ON, Canada
Venue:
Advances in Applied Mathematics
Year:
1994

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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 various sequences associated with this polynomial have been conjectured to be unimodal and indeed, log concave. We show that for any graph G, there is a subdivision for which the log concavity conjectures all hold. Further, we provide evidence for the well-known conjecture of the log concavity of the independent set numbers of a matroid.