Finding the most vital edges with respect to the number of spanning trees

A most vital edge of a graph (w.r.t. the spanning trees) is an edge whose deletion most drastically decreases the number of spanning trees. We present an algorithm for determining the most vital edges based on Kirchoff´s matrix-tree theorem whose asymptotic time-complexity can be reduced to that of the fastest matrix multiplication routine, currently O(n^2.376). The foundation for this approach is a more general algorithm for directed graphs for counting the rooted spanning arborescences containing each of the arcs of a digraph. A network can be modeled as a probabilistic graph. Under one such model proposed by Kel´mans, the all-terminal network reliability, maximizing the number of spanning trees is critical to maximizing reliability when edges are very unreliable. For this model, the most vital edges characterize the locations where an improvement of the reliability of the link most improves the reliability of the network