Some results on numerical ranges and factorizations of matrix polynomials

Some new factorization theorems for monic matrix polynomials are obtained. These theorems are based on the numerical range having the number of connected components equal to the degree of the polynomial. For second degree polynomials, sufficient conditions are given for the numerical range to have two connected components.     

The block numerical range of matrix polynomials

Block numerical ranges of matrix polynomials, especially the quadratic numerical range, are considered. The main results concern spectral inclusion, boundedness of the block numerical range, an estimate of the resolvent in terms of the quadratic numerical range, geometrical properties of the quadratic numerical range, and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions. As an application, we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials.     

On the spectral radius of tricyclic graphs with a fixed diameter

A tricyclic graph is a connected graph with n vertices and n + 2 edges. In this article, we determine graphs with the largest spectral radius among the n-vertex tricyclic graphs with given diameter d.     

Unitarily invariant norms related to the numerical radius

We determine the maximum in the class of unitarily invariant norms ∥·∥ such that w(A) ? ∥A∥ for all n-square matrices A. Here w(A) denotes the numerical radius of A.     

On an upper bound of the spectral radius of graphs

We give some upper bounds for the spectral radius of bipartite graph and graph, which improve the result in Hong’s Paper [Y. Hong, J.-L. Shu, K. Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory Ser. B 81 (2001) 177-183].     

Asymptotics of the Bohr radius for polynomials of fixed degree

We obtain a characterization and conjecture asymptotics of the Bohr radius for the class of complex polynomials in one variable. Our work is based on the notion of bound-preserving operators.     

Improved upper and lower bounds for the spectral radius of digraphs

Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius ρ(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on ρ(G) are given. We show that some known bounds can be obtained from our bounds.     

On the spectral radius of a nonnegative centrosymmetric matrix

In this paper, we discuss the spectral radius of nonnegative centrosymmetric matrices. By using the centrosymmetric structure, we establish some estimations of the spectral radius.     

Containment regions for zeros of polynomials from numerical ranges of companion matrices

Containment regions for the zeros of a monic polynomial are given with the aid of results for containment regions for the numerical range of certain bordered diagonal matrices which are applied to different types of companion matrices of the polynomial.     

Extending the formula to calculate the spectral radius of an operator

In a Banach space, Gelfand's formula is used to find the spectral radius of a continuous linear operator. In this paper, we show another way to find the spectral radius of a bounded linear operator in a complete topological linear space. We also show that Gelfand's formula holds in a more general setting if we generalize the definition of the norm for a bounded linear operator.

     

Sharp upper bounds of the spectral radius of a graph

Let $G$ be a simple connected graph with $n$ vertices and $m$ edges. The spectral radius $\rho \left(G\right)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of $\rho \left(G\right)$ which improves some known bounds: If $\frac{(k?2)(k?3)}{2}\le m?n\le \frac{k(k?3)}{2}$, where $k(3\le k\le n)$ is an integer, then $\rho \left(G\right)\le \sqrt{2m?n?k+\frac{5}{2}+\sqrt{2m?2n+\frac{9}{4}}}.$The equality holds if and only if $G$ is a complete graph $K_n$ or $K_4?e$, where $K_4?e$ is the graph obtained from $K_4$ by deleting some edge $e$.