Abstract
Consider graphs on the vertex set V={1, 2, …, n}, 1 < n < ∞, in which the edge between vertices i and j occurs with probability pij=pji, O≤pij≤ 1, independently for all edges. Let P=(pij) be the n × n symmetric matrix of edge probabilities with pij=0, i=1, …, n. Let T be the random number of spanning trees. E(T\P) denotes the redundancy, i.e., the expected number of spanning trees in random graphs with edge probability matrix P. An explicit (determinantal) formula for the sensitivity of the redundancy to changes in any edge probability, namely, ∂E(T\P)/∂pij, shows that this sensitivity equals the redundancy of random graphs in which vertices i and j have been collapsed to a single vertex or are connected with probability 1. There is an analogous formula for directed graphs.
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