Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.

Let $G=(V,E)$, $V=\{v_1,v_2,...,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$, respectively. The signless Laplacian of $G$ is defined as $L^{+}=D+A$ and the normalized signless Laplacian matrix as ${ℒ}^{+}=D^{-1/2}{L}^{+}{D}^{-1/2}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r\left(G\right)={\gamma }_2^{+}/{\gamma }_n^{+}$ and $l\left(G\right)={\gamma }_2^{+}-{\gamma }_n^{+}$, where ${\gamma }_1^{+}\ge {\gamma }_2^{+}\ge \cdots \ge {\gamma }_n^{+}\ge 0$ are eigenvalues of ${ℒ}^{+}$. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads...

A graph is determined by its signless Laplacian spectrum if no other non-isomorphic graph has the same signless Laplacian spectrum (simply $G$ is $DQS$). Let $T(a,b,c)$ denote the $T$-shape tree obtained by identifying the end vertices of three paths $P_{a+2}$, $P_{b+2}$ and $P_{c+2}$. We prove that its all line graphs $ℒ\left(T\right(a,b,c\left)\right)$ except $ℒ\left(T\right(t,t,2t+1\left)\right)$ ($t\ge 1$) are $DQS$, and determine the graphs which have the same signless Laplacian spectrum as $ℒ\left(T\right(t,t,2t+1\left)\right)$. Let ${\mu }_1\left(G\right)$ be the maximum signless Laplacian eigenvalue of the graph $G$. We give the limit of ${\mu }_1\left(ℒ\left(T(a,b,c)\right)\right)$, too.

Given a real n×n matrix A, we make some conjectures and prove partial results about the range of the function that maps the n-tuple x into the entrywise kth power of the n-tuple Ax. This is of interest in the study of the Jacobian Conjecture.

The structure of the set of positive unital maps between M₂(ℂ) and Mₙ(ℂ) (n ≥ 3) is investigated. We proceed with the study of the "quantized" Choi matrix thus extending the methods of our previous paper [MM2]. In particular, we examine the quantized version of Størmer's extremality condition. Maps fulfilling this condition are characterized. To illustrate our approach, a careful analysis of Tang's maps is given.