Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian...

If $\left\{v_i\right\}$ and $\left\{w_j\right\}$ are two families of unitary bases for ${\mathbf{C}}^n$, and $\theta$ is a fixed number, let $V^n$ and $W^n$ be subspaces of ${\mathbf{C}}^n$ spanned by $[\theta ·n]$ vectors in $\left\{v_i\right\}$ and $\left\{w_j\right\}$ respectively. We study the angle between $V^n$ and $W^n$ as $n$ goes to infinity. We show that when $\left\{v_i\right\}$ and $\left\{w_j\right\}$ arise in certain arithmetically defined families, the angles between $V^n$ and $W^n$ may either tend to $0$ or be bounded away from zero, depending on the behavior of an associated eigenvalue problem.

Vengono discusse delle condizioni sufficienti affinchè una matrice reale $A$ delle dimensioni $n\times n$ sia diagonalmente (o $D$-) stabile. Esse includono delle ipotesi geometriche (condizioni degli ortanti), e un criterio che generalizza un criterio di Carlson. Inoltre si discute la $D$-stabilità robusta per le matrici reali delle dimensioni $4\times 4$

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix $A\in {ℝ}^{m\times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on $\mathrm{Tr}\left(A^{-1}\right)$ and $\mathrm{Tr}\left(A^{-2}\right)$ have attracted attention recently, because they can be computed in $O\left(m\right)$ operations when $A$ is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed...

We investigate a notion of relative operator entropy, which develops the theory started by J. I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341-348]. For two finite sequences A = (A₁,...,Aₙ) and B = (B₁,...,Bₙ) of positive operators acting on a Hilbert space, a real number q and an operator monotone function f we extend the concept of entropy by setting $S_q^f\left(A\right|B):={\sum }_{j=1}^n{A}_j^{1/2}{\left(A_j^{-1/2}{B}_j{A}_j^{-1/2}\right)}^{q}f\left(A_j^{-1/2}{B}_j{A}_j^{-1/2}\right)A_j^{1/2}$, and then give upper and lower bounds for $S_q^f\left(A\right|B)$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219-235] under...