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Bounds on the subdominant eigenvalue involving group inverses with applications to graphs

Stephen J. Kirkland;  Neumann, Michael;  Bryan L. Shader — 1998

Czechoslovak Mathematical Journal

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are λ 1 > λ 2 ≥ ... ≥ λ n . In this paper we derive several lower and upper bounds, in particular on λ 2 and λ n , but also, indirectly, on μ = max 2 ≤ i ≤ n | λ i | . The bounds are in terms of the diagonal entries of the group generalized inverse, Q # , of the singular and irreducible M-matrix Q = λ 1 I - A . Our starting point is a spectral resolution for Q # . We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected...

On a bound on algebraic connectivity: the case of equality

Stephen J. Kirkland;  Neumann, Michael;  Bryan L. Shader — 1998

Czechoslovak Mathematical Journal

In a recent paper the authors proposed a lower bound on 1 - λ i , where λ i , λ 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...

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