| Summary: | 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.
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