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

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are ₁₂ ... _n. In this paper we derive several lower and upper bounds, in particular on ₂ and _n , but also, indirectly, on . The bounds are in terms of the diagonal entries of the group generalized inverse, Q ^#, of the singular and irreducible M-matrix Q = 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 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 ^#.