Let $A$ be an $n\times n$ symmetric, irreducible, and nonnegative matrix whose eigenvalues are $\lambda _1 > \lambda _2 \ge \ldots \ge \lambda _n$. In this paper we derive several lower and upper bounds, in particular on $\lambda _2$ and $\lambda _n$, but also, indirectly, on $\mu = \max _{2\le i \le n} |\lambda _i|$. The bounds are in terms of the diagonal entries of the group generalized inverse, $Q^{\#}$, of the singular and irreducible M-matrix $Q=\lambda _1 I-A$. 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 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 $L^{\#}$.
In this paper we obtain some results concerning the set ${\mathcal M} = \cup \bigl \lbrace \overline{R(\delta _A)}\cap \lbrace A\rbrace ^{\prime }\: A\in {\mathcal L(H)}\bigr \rbrace $, where $\overline{R(\delta _A)}$ is the closure in the norm topology of the range of the inner derivation $\delta _A$ defined by $\delta _A (X) = AX - XA.$ Here $\mathcal H$ stands for a Hilbert space and we prove that every compact operator in $\overline{R(\delta _A)}^w\cap \lbrace A^*\rbrace ^{\prime }$ is quasinilpotent if $A$ is dominant, where $\overline{R(\delta _A)}^w$ is the closure of the range of $\delta _A$ in the weak topology.
A method is presented making it possible to construct $po$-groups with a strong theory of quasi-divisors of finite character and with some prescribed properties as subgroups of restricted Hahn groups $H(\Delta ,\mathbb{Z})$, where $\Delta $ are finitely atomic root systems. Some examples of these constructions are presented.
A tree is classified as being type I provided that there are two or more Perron branches at its characteristic vertex. The question arises as to how one might construct such a tree in which the Perron branches at the characteristic vertex are not isomorphic. Motivated by an example of Grone and Merris, we produce a large class of such trees, and show how to construct others from them. We also investigate some of the properties of a subclass of these trees. Throughout, we exploit connections between characteristic vertices, algebraic connectivity, and Perron values of certain positive matrices associated with the tree.