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In a recent paper the authors proposed a lower bound on 1 - ?i, where ?i ? 1, is an eigenvalue of a transition matrix T of an ergodic Markov chain. The bound which involved the group inverse of I - T, was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when the graph is a weighted tree. It is shown that the bound is sharp only for certain Type I trees. Our proof involves characterizing the case of equality in an upper estimate for certain inner products due to A. Paz.

On a bound on algebraic connectivity: the case of equality