On a bound on algebraic connectivity: the case of equality
Czechoslovak Mathematical Journal, Tome 48 (1998) no. 1, pp. 65-76.

Voir la notice de l'article dans Czech Digital Mathematics Library

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 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.
Classification : 05C40, 05C50, 15A09, 15A42, 15A45, 15A51, 60J10 
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     title = {On a bound on algebraic connectivity: the case of equality},
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Kirkland, Stephen J.; Neumann, Michael; Shader, Bryan L. On a bound on algebraic connectivity: the case of equality. Czechoslovak Mathematical Journal, Tome 48 (1998) no. 1, pp. 65-76. https://geodesic-test.mathdoc.fr/item/CMJ_1998__48_1_a5/