Abstract
The number of spanning connected unicyclic subgraphs is a critical quantity to characterize the reliability of networks. In 1993, Colbourn in [Colbourn (1993)] proposed a problem about reliability polynomial: Can the number of spanning connected unicyclic subgraphs of a graph be computed efficiently? Up to now, there is no research about this problem. In this paper, we mainly study the entropy and the enumeration of spanning connected unicyclic subgraphs in self-similar networks. We propose a linear algorithm for computing the number of spanning connected unicyclic subgraphs in self-similar network (x, y)-flower. By the algorithm, the exact number of spanning connected unicyclic subgraphs is determined in (x, y)-flower networks. In addition, we define the entropy of spanning connected unicyclic subgraphs and obtain the entropies of spanning connected unicyclic subgraphs in (x, y)-flower networks.
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