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We consider problems of network reliability: the two-terminal network reliability consists, given an undirected graph G = (V, E), and a series of independent edge failure events, in computing the probability that two nodes remain connected. The all-terminal network reliability is the probability that the whole network remains connected. We present in the following two different approaches to compute two-terminal and all-terminal reliability, with various characteristics on the precision level of the result. We give an exact algorithm to compute the reliability in O(IVIf(w)2 + IEIf(w)) with f(x) = -X e (1+o(1))x and w is the tree-width of G. We also present Inx polynomial methods to give bounds on the reliability. We discuss methods to optimize the mean time to repair of the components.

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In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for Fp, where p is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, m and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.

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