### Nonexistence results for Hadamard-like matrices.

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Let $A$ be an $n\times n$ symmetric, irreducible, and nonnegative matrix whose eigenvalues are ${\lambda}_{1}>{\lambda}_{2}\ge ...\ge {\lambda}_{n}$. In this paper we derive several lower and upper bounds, in particular on ${\lambda}_{2}$ and ${\lambda}_{n}$, but also, indirectly, on $\mu ={max}_{2\le i\le n}\left|{\lambda}_{i}\right|$. The bounds are in terms of the diagonal entries of the group generalized inverse, ${Q}^{\#}$, of the singular and irreducible M-matrix $Q={\lambda}_{1}I-A$. 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...

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...

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