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A tree T with exactly one node u of degree larger than two is called a starlike tree and u is called the root node of T. Let S T n be a family of graphs consists of all starlike trees and the path with n + 1 nodes. For a fixed connected graph H with at least one edge, we construct a family of graphs F H (v) , n = { H (v) • T | T ∈ S T n } , where H (v) • T denotes the graph obtained by identifying some node, say v , of H with the root node of T. In this paper, we give an ordering of graphs in F H (v) , n by spectral radii coincides with the shortlex ordering of nondecreasing sequences of their branch lengths. • The spectral radius of the graph powerfully characterizes dynamic processes on networks, such as virus spread and synchronization. • In this paper, we give an ordering of graphs in F H (v) , n by spectral radii coincides with the shortlex ordering of nondecreasing sequences of their branch lengths. • The result generalizes the main result of Oliveira et al. in [10]. [ABSTRACT FROM AUTHOR]

In this paper, the problem of robust stability and robust stabilizability for periodically switched linear systems is studied. The stability radius involving structured perturbations acting on both coefficient matrices and switching moments is introduced and investigated. Some lower bounds for stability radii of periodically switched linear systems are provided. After that, we derive the notion of fast and slow stabilizability for these systems. Some characterizations for robust stabilizability under structured perturbations are established. Several examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The k th power of a graph G , denoted by Gk , is a graph with the same set of vertices as G such that two vertices are adjacent in Gk if and only if their distance in G is at most k. In this paper, we give upper and lower bounds on the A α -spectral radius of G 2. Furthermore, the first three largest A α -spectral radius of the second power of trees are determined. [ABSTRACT FROM AUTHOR]

Reversibility of cellular automata (CA) has been an extensively studied problem from both a theoretical and a practical point of view. It is known when a (2 R + 1) -cyclic cellular automaton with periodic boundary conditions (p.b.c.) is reversible (see Siap et al., 2013) but, as far as we know, no explicit expression is given for its inverse cellular automaton apart from the case R = 1 (see Encinas and del Rey, 2007). In this paper we give a closed formula for the inverse rule of a reversible (2 R + 1) -cyclic cellular automaton with p.b.c. over the finite field F 2 for any value of the neighbourhood radius R. It turns out that the inverse of a reversible (2 R + 1) -cyclic CA with p.b.c. is again a cyclic CA with p.b.c., but with a different neighbourhood radius, and this radius depends on certain numbers which need to be computed by a new algorithm we introduce. Finally, we apply our results to the case R = 1 (which is the ECA with Wolfram rule number 150) to introduce an alternative and improved expression for the inverse transition dipolynomial formulated in Encinas and del Rey (2007). We also illustrate these results by giving explicit computations for the inverse transition dipolynomial of a reversible cellular automaton with penta-cyclic rule. [ABSTRACT FROM AUTHOR]

Let consider A p denoting a class of analytical functions defined as f (z) = z p + a p + 1 z p + 1 + ⋯ + a p + n z p + n + ⋯ and p -valent in unit disc U = { z | | z | < 1 }. f (z) ∈ A p is expressed to be p -valently starlike in U if there is a positive figure ρ fulfilling ρ < | z | < 1, R e (z f ′ (z) f (z) ) > 0 , and ∫ 0 2 π R e (z f ′ (z) f (z) ) d θ = 2 p π , z = r e i θ , ρ < r < 1. Let us consider S *(p) denoting the family of f (z) in A p , being regular and p -valently starlike in U. It was proved by Goodman [3] that f (z) ∈ S *(p) is at most p -valent in U. In present study, some results about radius of starlikeness of p -valent λ -fractional operator were obtained. Also some relevant corollaries were given. Finally, a result associated with p -valent λ -fractional operator by using convolution was given as a conclusion. The aim of this study is to give some results on λ -fractional operator of f (z) ∈ S *(p). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish some spectral conditions for a balanced bipartite graph with large minimum degree to be traceable or Hamiltonian. [ABSTRACT FROM AUTHOR]

Abstract The integral funnel of the closed ball of the space L p , p > 1, with radius r and centered at the origin under Urysohn type integral operator is defined as the set of graphs of the images of all functions from given ball. Approximation of the integral funnel is considered. The closed ball of the space L p , p > 1, with radius r is replaced by the set consisting of a finite number of functions. The Hausdorff distance between integral funnel and the set consisting of the sections of the graphs of images of a finite number of functions is evaluated. It is proved that in the case of appropriate choosing of the discretization parameters the approximating sets converges to the integral funnel. [ABSTRACT FROM AUTHOR]

Abstract Let G be a graph of order n and let d i be the degree of the vertex v i in G for i = 1 , 2 , … , n. The weighted adjacency matrix A db of G is defined so that its (i, j)-entry is equal to d i + d j d i d j if the vertices v i and v j are adjacent, and 0 otherwise. The spectral radius ϱ 1 and the energy E d b of the A db -matrix are examined. Lower and upper bounds on ϱ 1 and E d b are obtained, and the respective extremal graphs are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we give upper bounds on Z t -spectral radius of a tensor A ( t = 1 , 2 ) , which extend the upper bounds of Brauer to tensors. Moreover, an upper bound on the Z 1 -spectral radius is proposed via modulus sum of the entries of certain dimension of A , which improves the upper bound given by Li et al. Numerical experiments are given to illustrate the utility of the upper bound. [ABSTRACT FROM AUTHOR]

The unique graphs with maximum spectral radius and signless Laplacian spectral radius are determined among trees, unicyclic graphs and bicyclic graphs respectively with fixed numbers of vertices and branch vertices (vertices of degree at least 3) using a unified approach. [ABSTRACT FROM AUTHOR]

Let ρ ( D ( G )) denote the distance spectral radius of a graph G and ∂ ( G → ) denote the distance signless Laplacian spectral radius of a digraph G → . Let G n , k D be the set of all k -connected graphs of order n with diameter D . In this paper, we first determine the unique graph with minimum distance spectral radius in G n , k D ; we then give sharp upper and lower bounds for the distance signless Laplacian spectral radius of strongly connected digraphs; we also characterize the digraphs having the maximal and minimal distance signless Laplacian spectral radii among all strongly connected digraphs; furthermore, we determine the extremal digraph with the minimal distance signless Laplacian spectral radius with given dichromatic number. [ABSTRACT FROM AUTHOR]

• Multidimensional mobility induces rich synergistic phenomena with network reciprocity, which depends on the cost of mobility. • The synergistic effects of mobility superposed upon network reciprocity are best expressed for small flow rates. • The increasing of radius of mobility may weaken network reciprocity. Collective cooperation and social mobility are ubiquitous in human societies. Due to information sharing and the complexity of everyday life, multidimensional mobility is also common, for example when moving into different districts of a city for better education or when settling permanently abroad due to better job prospects. Nevertheless, it is not clear how such complex mobility might affect cooperation in situations that constitute social dilemmas, where individual and public interests are at odds. Here, as an initial step to understand the impact of multidimensional mobility, we investigate one of its significant dimensions, namely the range of mobility. We propose an updating algorithm where individuals either move adaptively in the area bounded by a mobility radius or stay put for social learning. We use this on the prisoner's dilemma and the snowdrift game, and we find that an increase in either probability or radius of mobility may weaken network reciprocity, simply by decreasing the odds of meeting old interaction partners. However, if mobility is free, there is a window of parameters where synergies with network reciprocity are possible, and where indeed cooperation can be robust and significantly elevated. Local mobility in particular may favorably affect cooperation. In fact, even if mobility is costly, the failure of local mobility can often be associated with the risk caused by the shortage of available empty sites. We also find that the synergistic effects of mobility superposed upon network reciprocity are best expressed for small flow rates. Overall, we hope that our research will promote the better understanding of the complex interplay between networks reciprocity and mobility and their coaction. [ABSTRACT FROM AUTHOR]

Quality and size of mesh elements are important for optimizing the accuracy and convergence of mesh-based simulation processes. Often, a priori information, like internal material properties, of regions of interest is available, which can be used to locally specify the mesh element size for finding a good balance between the mesh resolution on the one hand and the runtime and memory performance on the other. In many applications, like the optimization of geometric parameters, multiple meshes of similar objects are required. Typical mesh element size specification methods, like scalar fields, are inflexible because of their dependence on the geometry of the object. To avoid the creation of a mesh element size specifications for each object manually, a specification method based on the objects topology rather than on its geometry, is needed. We tackle this problem by extending our meshing software ViennaMesh with a dynamic framework for locally specifying the size of mesh elements. Our approach aims for convenient utilization by using a XML-based configuration with support for arithmetic expressions. To achieve a high level of flexibility and reusability, this configuration can be specified based on the object’s topology, for example interfaces between different material regions. Additionally, geometric parameters, like the radius of the circumsphere of the object, are provided and can be used to, e.g., scale the local mesh element size according to the total size of the object. As a result, our configuration method is invariant under large set transformations, especially deformations, of the object enabling a high level of geometry independence. We depict the practicability of our approach by providing examples for meshes generated with this element sizing framework and discussing a geometry optimization application. [ABSTRACT FROM AUTHOR]

The class of bounded radius of rotation is generalization of the convex functions. The concept of functions bounded boundary rotation originated from Loewner (1917). But he did not use the present terminology. It was Paatero (1931, 1933) who systematically developed their properties and made an exhaustive study of the class V k . In the present paper we will investigate the class of harmonic mappings which is related to the class of bounded boundary rotation. [ABSTRACT FROM AUTHOR]

Recently, a new treatment based on Taylor’s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented. It has been successfully applied to enlarge the convergence radius of the modified Newton’s method and Osada’s method for multiple roots. This paper re-investigates the convergence radius of Halley’s method under the condition that the derivative f ( m + 1 ) of function f satisfies the center-Hölder continuous condition. We show that our result can be obtained under much weaker condition and has a wider range of application than that given by Bi et. al.(2011) [21]. [ABSTRACT FROM AUTHOR]

Recently, a new treatment based on Taylor's expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented. It has been successfully applied to enlarge the estimate of the convergence radius of the modified Newton's method for multiple roots. Using the similarly treatment, this paper investigates the convergence radius of the Osada's method under the condition that the derivative f(m+1) of function f satisfies the center-Hölder continuous condition. By some examples, we show the treatment is simpler and efficient once again. The uniqueness ball of solution is also discussed. [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple connected weighted graph on n vertices, in which the edge weights are positive definite matrices. The eigenvalues of G are the eigenvalues of its adjacency matrix. In this note, we present a correction in equality part in Theorem 2 [S. Sorgun, S. Büyükköse, The new upper bounds on the spectral radius of weighted graphs, Appl. Math. Comput. 218 (2012) 5231–5238]. In addition, some related results are also provided. [ABSTRACT FROM AUTHOR]

Abstract: Let be a given interval parametric matrix (the set of all parametric matrices when the m-dimensional parameter vector p varies within a given interval vector ) whose entries depend affine linearly on p. Also, let L denote the set of all real eigenvalues of the bundle . In this paper, first the concept of regularity radius of the interval parametric matrix is introduced. It is then shown that there exists certain relationship between the problem of establishing if belongs to L or not and the numerical value of the regularity radius of the interval parametric matrix . The results presented may be useful in designing a method for determining or assessing the set L. [Copyright &y& Elsevier]

Abstract: The estimation of zeros of a polynomial have been done by many mathematicians over the years using various approaches. In this paper we estimate the upper bound for the zeros of a given polynomial using Hilbert space technique involving Frobenius companion matrix and numerical radius. We first obtain numerical range and numerical radius for certain class of matrices and use them to estimate the bounds for zeros of a given polynomial. We illustrate with examples to show that the estimations obtained here is better than the previously known estimations. We also obtain a sequence of real numbers which converges exactly to the spectral radius of some special class of matrices. [Copyright &y& Elsevier]

Abstract: In this paper, we present preconditioned generalized accelerated overrelaxation methods. We compare the spectral radii of the iteration matrices of the preconditioned and original methods. The comparison results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. Finally, we present two numerical examples to confirm our theoretical results. [Copyright &y& Elsevier]

Abstract: Texture plays an important role in computer vision. It is one of the most important visual attributes used in image analysis, once it provides information about pixel organization at different regions of the image. This paper presents a novel approach for texture characterization, based on complexity analysis. The proposed approach expands the idea of the Mass–radius fractal dimension, a method originally developed for shape analysis, to a set of coordinates in 3D-space that represents the texture under analysis in a signature able to characterize efficiently different texture classes in terms of complexity. An experiment using images from the Brodatz album illustrates the method performance. [Copyright &y& Elsevier]

Abstract: In this paper, we use a technique introduced in the paper [P. Dankelmann, R.C. Entringer, Average distance, minimum degree and spanning trees, J. Graph Theory 33 (2000), 1–13] to obtain a strengthening of an old classical theorem by Erdös et al. [P. Erdös, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory B 47 (1989), 73–79] on radius and minimum degree. To be more detailed, we will prove that if G is a connected graph of order n with the minimum degree δ, then the radius G does not exceedwhere t is the irregularity index (that is the number of distinct terms of the degree sequence of G) which has been recently defined in the paper [S. Mukwembi, A note on diameter and the degree sequence of a graph, Appl. Math. Lett. 25 (2012), 175–178]. We claim that our result represent the tightest bound that ever been obtained until now. [Copyright &y& Elsevier]

Abstract: Let be a simple connected graph. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of G is . In this paper, we obtain some new and improved sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph G. [Copyright &y& Elsevier]

Abstract: Let denote the class of all analytic functions f in the unit disc with the normalization and satisfying the conditionthus is in the interior of the right half of the lemniscate of Bernoulli . The relations between and others classes geometrically defined are considered. The radii of -convexity, of -starlikeness (and some of others) of are determined. [Copyright &y& Elsevier]

• The paper characterizes the robust second-moment stability of stochastic linear systems subject to varying delays. • The stability condition, difficult to check, is converted into a randomized test. • The usefulness of this approach was illustrated through a real-time electronic application (the paper shows real-time data collected in a laboratory). This paper characterizes the robust second-moment stability of stochastic linear systems subject to varying delays. The delays assume a particular form suitable to represent packet loss in networked control systems, under the zero-order hold feedback. The proposed robust stability condition requires checking the spectral radius of an appropriate matrix that that depends on uncertain parameters belonging to a polytope. Due to this polytope's dependence, checking the spectral radius is difficult from the numerical viewpoint. As an attempt to solve the problem, we convert the polytope-based condition into a randomized approach. Namely, we present probability bounds that help us certificate the robust second-moment stability under high probability. A real-time electronic application illustrates the potential benefits of our approach. [ABSTRACT FROM AUTHOR]

Abstract: Estimates for the computation of the joint spectral radius defined by a bounded collection of square matrices and of same size will be given. In particular, the computational result shows that for matrices constructed by four-coefficient two-scale dilation equation our approach leads to the exact value of this joint spectral radius. [Copyright &y& Elsevier]

• The α -spectral radius of general hypergraphs is introduced and investigated. • New upper bounds about the α -spectral radius for nonregular f-connected general linear hypergraph and proper subhypergraphs of regular f-connected general hypergraph are established. For a connected hypergraph G with adjacency tensor A (G) and degree diagonal tensor D (G) , and any real α satisfying 0 ≤ α ≤ 1, define the tensor A α (G) as A α (G) = α D (G) + (1 − α) A (G). The largest modulus of the eigenvalues of A α (G) , is called the α -spectral radius of G , denoted by λ α (G). In this paper, we mainly obtain an upper bound of the α -spectral radius of a nonregular f -connected general linear hypergraph G. And we give an upper bound of the α -spectral radius of a subhypergraph of a d -regular f -connected general hypergraph. [ABSTRACT FROM AUTHOR]

Abstract: We obtain the small-amplitude expansion for the period of the nonlinear oscillator with the initial conditions and , calculate its radius of convergence and show that the inverted perturbation series appears to converge smoothly from below. [Copyright &y& Elsevier]

Let G be a connected graph. The greatest eigenvalue and the spread of the sum of the adjacency matrix and the distance matrix of G are called the adjacency-distance spectral radius and the adjacency-distance spread of G , respectively. Both quantities are used as molecular descriptors in chemoinformatics. We establish some properties for the adjacency-distance spectral radius and the adjacency-distance spread by proposing local grafting operations such that the adjacency-distance spectral radius is decreased or increased. Hence, we characterize those graphs that uniquely minimize and maximize the adjacency-distance spectral radii in several sets of graphs, and determine trees with small adjacency-distance spreads. It transpires that the adjacency-distance spectral radius satisfies the requirements of a branching index. [ABSTRACT FROM AUTHOR]

For a real number α ∈ [0, 1], the A α -matrix of a graph G is defined as A α (G) = α D (G) + (1 − α) A (G) , where A (G) and D (G) are the adjacency matrix and diagonal degree matrix of G , respectively. The A α -spectral radius of G , denoted by ρ α (G), is the largest eigenvalue of A α (G). In this paper, the Nordhaus–Gaddum type bounds for the A α -spectral radius are considered. [ABSTRACT FROM AUTHOR]

For any real α ∈ [0, 1], A α (G) = α D (G) + (1 − α) A (G) is the A α -matrix of a graph G , where A (G) is the adjacency matrix of G and D (G) is the diagonal matrix of the degrees of G. This paper presents some extremal results about the spectral radius λ 1 (A α (G)) of A α (G) that generalize previous results about λ 1 (A 0 (G)) and λ 1 (A 1 2 (G)). In this paper, we study the behavior of the A α -spectral radius under some graph transformations for α ∈ [0, 1). As applications, we show that the greedy tree has the maximum A α -spectral radius in G D when D is a tree degree sequence firstly. Furthermore, we determine that the greedy unicyclic graph has the largest A α -spectral radius in G D when D is a unicyclic graphic sequence, where G D = { G ∣ G is connected with D as its degree sequence }. [ABSTRACT FROM AUTHOR]