### On a bound on algebraic connectivity: the case of equality

In a recent paper the authors proposed a lower bound on $1-{\lambda}_{i}$, where ${\lambda}_{i}$, ${\lambda}_{i}\ne 1$, 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...