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

Abstract

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are λ₁〉λ₂≥ ... ≥λ_n. In this paper we derive several lower and upper bounds, in particular on λ₂ and λ _n , but also, indirectly, on $\mu = \mathop {\max }\limits_{2 \leqslant i \leqslant n} |\lambda _i |$. The bounds are in terms of the diagonal entries of the group generalized inverse, Q ^#, of the singular and irreducible M-matrix Q = λ₁ 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 graphs, where now Q becomes L, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of L ^# and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of L, by exploiting the diagonal entries of L ^#.