On Network Reliability
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The all terminal reliability of a graph G is the probability that at least a spanning tree is operational, given that vertices are always operational and edges independently operate with probability p in [0,1]. In this thesis, an investigation of all terminal reliability is undertaken. An open problem regarding the non-existence of optimal graphs is settled and analytic properties, such as roots, thresholds, inflection points, fixed points and the average value of the all terminal reliability polynomial on [0,1] are studied. A new reliability problem, the k -clique reliability for a graph G is introduced. The k-clique reliability is the probability that at least a clique of size k is operational, given that vertices operate independently with probability p in [0,1] . For k-clique reliability the existence of optimal networks, analytic properties, associated complexes and the roots are studied. Applications to problems regarding independence polynomials are developed as well.