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Sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix

You, Lihua; Shu, Yujie; Yuan, Pingzhi
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/07/2015
Relevância na Pesquisa
36.09%
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph.; Comment: 19 pages

Application of topological radicals to calculation of joint spectral radii

Shulman, Victor S.; Turovskii, Yuri V.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 02/05/2008
Relevância na Pesquisa
46.03%
It is shown that the joint spectral radius $\rho(M)$ of a precompact family $M$ of operators on a Banach space $X$ is equal to the maximum of two numbers: the joint spectral radius $\rho_{e}(M)$ of the image of $M$ in the Calkin algebra and the Berger-Wang radius $r(M)$ defined by the formula \[ r(M)=\underset{n\to\infty}{\limsup}(\sup\left\{\rho(a):a\in M^{n}\right\} ^{1/n}) . \] Some more general Banach-algebraic results of this kind are also proved. The proofs are based on the study of special radicals on the class of Banach algebras.

Lower bounds of the skew spectral radii and skew energy of oriented graphs

Chen, Xiaolin; Li, Xueliang; Lian, Huishu
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.27%
Let $G$ be a graph with maximum degree $\Delta$, and let $G^{\sigma}$ be an oriented graph of $G$ with skew adjacency matrix $S(G^{\sigma})$. The skew spectral radius $\rho_s(G^{\sigma})$ of $G^\sigma$ is defined as the spectral radius of $S(G^\sigma)$. The skew spectral radius has been studied, but only few results about its lower bound are known. This paper determines some lower bounds of the skew spectral radius, and then studies the oriented graphs whose skew spectral radii attain the lower bound $\sqrt{\Delta}$. Moreover, we apply the skew spectral radius to the skew energy of oriented graphs, which is defined as the sum of the norms of all the eigenvalues of $S(G^\sigma)$, and denoted by $\mathcal{E}_s(G^\sigma)$. As results, we obtain some lower bounds of the skew energy, which improve the known lower bound obtained by Adiga et al.; Comment: 16 pages, 2 figures

Laplacian and signless Laplacian spectral radii of graphs with fixed domination number

Xing, Rundan; Zhou, Bo
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 28/10/2013
Relevância na Pesquisa
46.1%
In this paper, we determine the maximal Laplacian and signless Laplacian spectral radii for graphs with fixed number of vertices and domination number, and characterize the extremal graphs respectively.

Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

Kozyakin, Victor
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.3%
Recently Blondel, Nesterov and Protasov proved that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative earlier used by the author and his companions to analyze the minimax relations between the spectral radii of matrix products. Axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property, which includes the sets of non-negative matrices with independent row uncertainty. This class of matrices, supplemented by the zero and identity matrices, forms a semiring with the Minkowski operations of addition and multiplication of matrix sets, which gives means to construct new sets of non-negative matrices possessing the finiteness property for the generalized and the lower spectral radii.; Comment: 17 pages, 38 bibliography references, minor corrections, accepted for publication in Linear Algebra and its Applications

The extremal spectral radii of $k$-uniform supertrees

Li, Honghai; Shao, Jiayu; Qi, Liqun
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 28/05/2014
Relevância na Pesquisa
56.54%
In this paper, we study some extremal problems of three kinds of spectral radii of $k$-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidence $Q$-spectral radius). We call a connected and acyclic $k$-uniform hypergraph a supertree. We introduce the operation of "moving edges" for hypergraphs, together with the two special cases of this operation: the edge-releasing operation and the total grafting operation. By studying the perturbation of these kinds of spectral radii of hypergraphs under these operations, we prove that for all these three kinds of spectral radii, the hyperstar $\mathcal{S}_{n,k}$ attains uniquely the maximum spectral radius among all $k$-uniform supertrees on $n$ vertices. We also determine the unique $k$-uniform supertree on $n$ vertices with the second largest spectral radius (for these three kinds of spectral radii). We also prove that for all these three kinds of spectral radii, the loose path $\mathcal{P}_{n,k}$ attains uniquely the minimum spectral radius among all $k$-th power hypertrees of $n$ vertices. Some bounds on the incidence $Q$-spectral radius are given. The relation between the incidence $Q$-spectral radius and the spectral radius of the matrix product of the incidence matrix and its transpose is discussed.

Lower bounds for the spectral radii of adjacency operators on Baumslag-Solitar groups

Dykema, Ken; Redelmeier, Daniel
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.1%
We will use free probability techniques to find lower bounds for the spectral radii of the adjacency operators on the Caley graphs of some non-amenable Baumslag-Solitar groups with the standard generators.; Comment: The preprint is now withdrawn, because of an error in the technique. (Certain words were counted twice.) However, we are working on producing a correct argument and we are able to obtain a nontrivial lower bound

Counting Spectral Radii of Matrices with Positive Entries

da Silva, J. A. Dias; Freitas, Pedro J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 06/05/2013
Relevância na Pesquisa
46.1%
The sum-product conjecture of Erd\H os and Szemer\'edi states that, given a finite set $A$ of positive numbers, one can find asymptotic lower bounds for $\max\{|A+A|,|A\cdot A|\}$ of the order of $|A|^{1+\delta}$ for every $\delta <1$. In this paper we consider the set of all spectral radii of $n\times n$ matrices with entries in $A$, and find lower bounds for the cardinality of this set. In the case $n=2$, this cardinality is necessarily larger than $\max\{|A+A|,|A\cdot A|\}$.

Spectral Radii of Bounded Operators on Topological Vector Spaces

Troitsky, Vladimir G.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 08/04/2000
Relevância na Pesquisa
56.25%
In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several important differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern.; Comment: 36 pages

Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs

Najar, Hatem; Olendski, Oleg
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.1%
Bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$ are investigated. We impose the Neumann boundary condition on the two concentric windows of the radii $a$ and $ b$ located on the opposite walls and the Dirichlet boundary condition on the remaining part of the boundary of the strip. We prove that such a system exhibits discrete eigenvalues below the essential spectrum for any $a,b>0$. When $a$ and $b$ tend to the infinity, the asymptotic of the eigenvalue is derived. A comparative analysis with the one-window case reveals that due to the additional possibility of the regulating energy spectrum the anticrossing structure builds up as a function of the inner radius with its sharpness increasing for the larger outer radius. Mathematical and physical interpretation of the obtained results is presented; namely, it is derived that the anticrossings are accompanied by the drastic changes of the wave function localization. Parallels are drawn to the other structures exhibiting similar phenomena; in particular, it is proved that, contrary to the two-dimensional geometry, at the critical Neumann radii true bound states exist.; Comment: 25 pages, 7 figures

An algorithm computing non solvable spectral radii of $p$-adic differential equations

Pulita, Andrea
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/01/2013
Relevância na Pesquisa
46.1%
We obtain an algorithm computing explicitly the values of the non solvable spectral radii of convergence of the solutions of a differential module over a point of type 2, 3 or 4 of the Berkovich affine line.; Comment: 5 pages

Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces

Thieme, Horst R
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 25/06/2014
Relevância na Pesquisa
46.1%
It is well known that an ordered normed vector space $X$ with normal cone $X_+$ has an order-preserving norm that is equivalent to the original norm. Such an equivalent order-preserving norm is given by \begin{equation} \sharp x \sharp = \max \{ d(x, X_+), d(x, - X_+)\}, \qquad x \in X. \end{equation} This paper explores the properties of this norm and of the half-norm $\psi(x) = d(x,-X_+)$ independently of whether or not the cone is normal. We use $\psi$ to derive comparison principles for the solutions of abstract integral equations, derive conditions for point-dissipativity of nonlinear positive maps, compare Collatz-Wielandt numbers, bounds, and order spectral radii for bounded homogeneous maps and give conditions for a local upper Collatz-Wielandt radius to have a lower positive eigenvector.

On spectral radius of strongly connected digraphs

Li, Jianping; Zhou, Bo
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 01/05/2013
Relevância na Pesquisa
36.25%
We determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $n\ge 4$, and thus we answer affirmatively the problem whether the unique digraph which achieves the minimum spectral radius among all strongly connected bicyclic digraphs of order $n$ achieves the second minimum spectral radius among all strongly connected digraphs of order $n$ for $n\ge 4$ proposed in [H. Lin, J. Shu, A note on the spectral characterization of strongly connected bicyclic digraphs, Linear Algebra Appl. 436 (2012) 2524--2530]. We also discuss the strongly connected bicyclic digraphs with small and large spectral radii respectively.

Some Spectral Properties of Odd-Bipartite $Z$-Tensors and Their Absolute Tensors

Chen, Haibin; Qi, Liqun
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/03/2015
Relevância na Pesquisa
45.99%
Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly even-bipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a $Z$-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that $Z$-tensor. When the order is even and the $Z$-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the $Z$-tensor and the largest H-eigenvalue of the absolute tensor of that $Z$-tensor are equal, if and only if the $Z$-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric $Z$-tensor with nonnegative diagonal entries and the absolute tensor of the $Z$-tensor are diagonal similar, if and only if the $Z$-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric $Z$-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the $Z$-tensor and the spectrum of absolute tensor of that $Z$-tensor...

The Berger-Wang formula for the Markovian joint spectral radius

Kozyakin, Victor
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.3%
The Berger-Wang formula establishes equality between the joint and generalized spectral radii of a set of matrices. For matrix products whose multipliers are applied not arbitrarily but in accordance with some Markovian law, there are also known analogs of the joint and generalized spectral radii. However, the known proofs of the Berger-Wang formula hardly can be directly applied in the case of Markovian products of matrices since they essentially rely on the arbitrariness of appearance of different matrices in the related matrix products. Nevertheless, as has been shown by X. Dai the Berger-Wang formula is valid for the case of Markovian analogs of the joint and the generalized spectral radii too, although the proof in this case heavily exploits the more involved techniques of multiplicative ergodic theory. In the paper we propose a matrix theory construction allowing to deduce the Markovian analog of the Berger-Wang formula from the classical Berger-Wang formula.; Comment: 13 pages, 29 bibliography references; minor corrections; accepted for publication in Linear Algebra and its Applications

On the spectral radii of bicyclic graphs with fixed independence number

Yuan, Xiying
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/02/2014
Relevância na Pesquisa
46.1%
Bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. In this paper, we determine the graph which alone maximizes the spectral radii among all the bicyclic graphs on $n$ vertices with fixed independence number.; Comment: 13pages

A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones

Akian, Marianne; Gaubert, Stephane; Nussbaum, Roger
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.05%
Several notions of spectral radius arise in the study of nonlinear order-preserving positively homogeneous self-maps of cones in Banach spaces. We give conditions that guarantee that all these notions lead to the same value. In particular, we give a Collatz-Wielandt type formula, which characterizes the growth rate of the orbits in terms of eigenvectors in the closed cone or super-eigenvectors in the interior of the cone. This characterization holds when the cone is normal and when a quasi-compactness condition, involving an essential spectral radius defined in terms of $k$-set-contractions, is satisfied. Some fixed point theorems for non-linear maps on cones are derived as intermediate results. We finally apply these results to show that non-linear spectral radii commute with respect to suprema and infima of families of order preserving maps satisfying selection properties.; Comment: 24 pages, v2: minor improvements and updated references

Mather sets for sequences of matrices and applications to the study of joint spectral radii

Morris, Ian D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/09/2011
Relevância na Pesquisa
46.16%
The joint spectral radius of a compact set of d-times-d matrices is defined ?to be the maximum possible exponential growth rate of products of matrices drawn from that set. In this article we investigate the ergodic-theoretic structure of those sequences of matrices drawn from a given set whose products grow at the maximum possible rate. This leads to a notion of Mather set for matrix sequences which is analogous to the Mather set in Lagrangian dynamics. We prove a structure theorem establishing the general properties of these Mather sets and describing the extent to which they characterise matrix sequences of maximum growth. We give applications of this theorem to the study of joint spectral radii and to the stability theory of discrete linear inclusions. These results rest on some general theorems on the structure of orbits of maximum growth for subadditive observations of dynamical systems, including an extension of the semi-uniform subadditive ergodic theorem of Schreiber, Sturman and Stark, and an extension of a noted lemma of Y. Peres. These theorems are presented in the appendix.

Maximizing spectral radii of uniform hypergraphs with few edges

Fan, Yi-Zheng; Tan, Ying-Ying; Peng, Xi-Xi; Liu, An-Hong
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.16%
In this paper we investigate the hypergraphs whose spectral radii attain the maximum among all uniform hypergraphs with given number of edges. In particular we characterize the hypergraph(s) with maximum spectral radius over all unicyclic hypergraphs, linear or power unicyclic hypergraphs with given girth, linear or power bicyclic hypergraphs, respectively.

Ordering uniform supertrees by their spectral radii

Yuan, Xiying
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/02/2015
Relevância na Pesquisa
46.35%
A connected and acyclic hypergraph is called a supertree. In this paper we mainly focus on the spectral radii of uniform supertrees. Li, Shao and Qi determined the first two $k$-uniform supertrees with large spectral radii among all the $k$-uniform supertrees on $n$ vertices [H. Li, J. Shao, L. Qi, The extremal spectral radii of $k$-uniform supertrees, arXiv:1405.7257v1, May 2014]. By applying the operation of moving edges on hypergraphs and using the weighted incidence matrix method we extend the above order to the fourth $k$-uniform supertree.; Comment: arXiv admin note: text overlap with arXiv:1405.7257 by other authors