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We introduce a formal framework to study the multiple unicast problem for a coded network in which the network code is linear over a finite field and fixed. We show that the problem corresponds to an interference alignment problem over a finite field. In this context, we establish an outer bound for the achievable rate region and provide examples of networks where the bound is sharp. We finally give evidence of the crucial role played by the field characteristic in the problem. [ABSTRACT FROM AUTHOR]

For r ∈ (0 , 1) , let E (r) be the complete elliptic integral of the second kind. In this paper, the authors obtain a accuracy approximation for E (r) , which improve the related inequalities satisfied by E (r) . [ABSTRACT FROM AUTHOR]

Let Z(G) be the zero forcing number of a simple connected graph G. In this paper, we study the relationship between the zero forcing number of a graph and its (normalized) Laplacian eigenvalues. We provide the upper and lower bounds on Z(G) in terms of its (normalized) Laplacian eigenvalues, respectively. Our bounds extend the existing bounds for regular graphs. [ABSTRACT FROM AUTHOR]

The homogeneous weight (metric) is useful in the construction of codes over a ring of integers Z p l (p prime and l ≥ 1 an integer). It becomes Hamming weight when the ring is taken to be a finite field and becomes Lee weight when the ring is taken to be Z 4 . This paper presents homogeneous weight distribution and total homogeneous weight of burst and repeated burst errors in the code space of n-tuples over Z p l . Necessary and sufficient conditions for existence of an (n, k) linear code over Z p l correcting the error patterns with respect to the homogeneous weight are derived. [ABSTRACT FROM AUTHOR]

Motivated by the potential connection between metric-affine gravity and linear generalized uncertainty principle (GUP) in the phase space, we develop a covariant form of linear GUP and an associated modified Poincaré algebra, which exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP. We use three-torus geometry to visually represent linear GUP within a covariant framework. The three-torus area provides an exact geometric representation of Bekenstein's universal bound. We depart from Bousso's approach, which adapts Bekenstein's bound by substituting the Schwarzschild radius ( r s ) with the radius (R) of the smallest sphere enclosing the physical system, thereby basing the covariant entropy bound on the sphere's area. Instead, our revised covariant entropy bound is described by the area of a three-torus, determined by both the inner radius r s and outer radius R where r s ⩽ R due to gravitational stability. This approach results in a more precise geometric representation of Bekenstein's bound, notably for larger systems where Bousso's bound is typically much larger than Bekensetin's universal bound. Furthermore, we derive an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the three-torus covariant entropy bound, suggesting a new avenue of quantum gravity. [ABSTRACT FROM AUTHOR]

In this paper, we explore some properties of the dual Wills functional, which are part of the dual Brunn–Minkowski theory. We give the upper and lower bounds for the dual Wills functional in terms of the 1-th dual volume of star bodies. Moreover, an inequality that is associated with the section of convex bodies for isotropic measures is presented. [ABSTRACT FROM AUTHOR]

The main goal of this note is to improve the best known bounds for the Stieltjes constants, using the method of steepest descent that was applied in 2011 by Coffey and Knessl in order to approximate these constants. [ABSTRACT FROM AUTHOR]

In this study, the Laplacian matrix concept for the power graph of a finite cyclic group is redefined by considering the block matrix structure. Then, with the help of the eigenvalues of the Laplacian matrix in question, the concept of Laplacian energy for the power graphs of finite cyclic groups was defined and introduced into the literature. In addition, boundary studies were carried out for the Laplacian energy in question using the concepts the trace of a matrix, the Cauchy-Schwarz inequality, the relationship between the arithmetic mean and geometric mean, and determinant. Later, various results were obtained for the Laplacian energy in question for cases where the order of a cyclic group is the positive integer power of a prime. [ABSTRACT FROM AUTHOR]

In the paper, the author presents lower and upper bounds for norms of the sine and cosine functions along a circle on the complex plane. [ABSTRACT FROM AUTHOR]

The second multiplicative Zagreb eccentricity index E*2 (G) of a simple connected graph G is expressed as the product of the weights ε(a)εG(b) over all edges ab of G, where ε(a) stands for the eccentricity of the vertex a in G. In this paper, some extremal problems on the E*2 index over some special graph classes including trees, unicyclic graphs and bicyclic graphs are examined, and the corresponding extremal graphs are characterized. Besides, the relationships between this vertex-eccentricity-based graph invariant and some well-known parameters of graphs and existing graph invariants such as the number of vertices, number of edges, minimum vertex degree, maximum vertex degree, eccentric connectivity index, connective eccentricity index, first multiplicative Zagreb eccentricity index and second multiplicative Zagreb index are investigated. [ABSTRACT FROM AUTHOR]

The extropy measure proposed by Lad, Sanfilippo, and Agro (Statistical Science 30 (1):40–58, 2015) as a dual function of Shannon entropy has received a considerable attention in the last five years. In this paper, we first show a representation for the weighted extropy of ranked set sampling in terms of quantile and density-quantile functions. Then we provide some related results including monotone properties, stochastic orders, characterizations and sharp bounds. Moreover, the so obtained results show how the weighted extropy of ranked set sampling compares with its counterpart of simple random sampling. [ABSTRACT FROM AUTHOR]

Globally, Diabetic mellitus (DM) is the most common chronic health problem worldwide. It elevates the blood glucose levels in patients, resulting in severe health problems like fatigue, numbness on feet or hands, blurred vision, etc., particularly, type-2 DM prevalence rate is increasing day by day. However, it is a curable disease, so it is vital to identify the disease early to reduce the complications. Correspondingly, traditional screening methods for DM include a blood test, blood sugar meter, etc. It is a costly and time-consuming method. Moreover, it requires physicians to process the test. To resolve this, numerous existing researches focused on the detection of DM with Artificial Intelligence but lacked accuracy and speed. Therefore, the proposed system used a particular set of procedures to enhance the classification performance. Initially, the data is preprocessed to remove the noise and enhance the quality of the data. Then, the feature extraction is carried out using Kernel Principle Component Analysis to take the significant features in the data. Further, feature selection is processed using Novel Bisection Of Branch And Bound Algorithm to select the important features. Formerly, classification was attained through Novel-based Radial basis function based Kohonen Self-Organizing Map with Grey Wolf Optimizer. The performance of the system is verified through performance metrics. The outcome of the analysis signifies the proposed model accomplished an accuracy of 97.07%. The respective system is planned to contribute to the research related to diabetes and to support qualified doctors in diabetes treatment. [ABSTRACT FROM AUTHOR]

This document presents a study on the power graphs of finite cyclic groups. The authors derive lower and upper bounds for the energies of these power graphs by analyzing the adjacency matrix structure. They also conduct boundary studies to compare the obtained bounds. The document includes references to related research on graph energy and power graphs of other mathematical structures. [Extracted from the article]

For a graph G, its atom-bond connectivity (ABC) index (respectively, atom-bond sum connectivity (ABS) index) is defined as the addition of the numbers... over all unordered pairs of adjacent vertices {vi, vj} of G, where di and dj denote the degrees of vi and vj, respectively. In this paper, sharp upper bounds on the ABC and ABS indices are derived. All the graphs that attain the obtained bounds are also completely characterized. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes defined in 2023. We show that it extends the family of j-maximum rank distance codes and contains the j-binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the zeta function for tensor codes. We establish connections between this object and the weight enumerator of a tensor code with respect to the Ravagnani-type anticodes. We introduce a new refinement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes classified in 2023 and we derive the corresponding MacWilliams identities. In this framework, we also define a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application, we derive connections on the tensor weights. [ABSTRACT FROM AUTHOR]

In this article, we study stochastic orders over an interval. Mainly, we focus on orders related to the Laplace transform. The results are then applied to obtain a bound for heavy-tailed distributions and are illustrated by some examples. We also indicate how these ordering relationships can be adapted to the classical risk model in order to derive a moment bound for ruin probability. Finally, we compare it with other existing bounds. [ABSTRACT FROM AUTHOR]

For a graph G with vertex set V G and edge set E G , the symmetric division deg index is defined as SDD (G) = ∑ u v ∈ E G (d u d v + d v d u) , where d u denotes the degree of vertex u in G. In 2018, Furtula et al. confirmed the quality of SDD index exceeds that of some more popular VDB indices, in particular that of the GA index. They show a close connection between the SDD index and the earlier well-established GA index. Thus, it is meaningful and important to consider the chemical and mathematical properties of the SDD index. In this paper, we determine some sharp bounds on the symmetric division deg index of graphs and line graphs and characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

The tensor eigenvalue problem has been widely studied in recent years. In this paper, we propose a new constructive way to estimate bounds on the H-eigenvalues of two kinds of interval tensors including nonnegative tensors and real symmetric tensors of even order. As applications, perturbation analysis of both nonnegative tensors and real symmetric tensors is numerically studied. One open question in Qi (Linear Algebra Appl 439:228–238, 2013) has also been answered. [ABSTRACT FROM AUTHOR]

This paper presents a cryptanalysis attack on the RSA variant with modulus $N=p^rq$ for $r\geq 2$ with three public and private exponents $(e_1,d_1),$ $(e_2,d_2),$ $(e_3,d_3)$ sharing the same modulus $N$ where $p$ and $q$ are consider to prime having the same bit size. Our attack shows that we get the private exponent $\sigma_1\sigma_2\sigma_3

Let p, q be two positive integers. The 3-graph F(p, q) is obtained from the complete 3-graph Kp3 by adding q new vertices and p (2 q) new edges of the form vxy for which v ∈ V(Kp3) and {x, y} are new vertices. It frequently appears in many literatures on the Turán number or Turán density of hypergraphs. In this paper, we first construct a new class of r-graphs which can be regarded as a generalization of the 3-graph F(p, q), and prove that these r-graphs have the same Turán density under some situations. Moreover, we investigate the Turán density of the F(p, q) for small p, q and obtain some new bounds on their Turán densities. [ABSTRACT FROM AUTHOR]

Visual matching of plane images has promoted the development of artificial intelligence and digital vision. High-precision visual matching can promote the innovation of geometric measurement, visual navigation and other fields. Therefore, a non-linear visual matching model with inherent constraints is established in this paper. First, according to the principle of visual imaging, a non-linear conversion model of visual point coordinates is proposed, and the deviation of coordinate points is proofread. Then, inherent boundary constraints are introduced into the model to improve the accuracy of visual matching. Finally, through analysis and evaluation of error, results are generated showing that the visual matching model can effectively solve the shortcoming of low-matching accuracy in feature points, and provide more accurate data support for 3D calculation of images.

This paper presents a cryptanalysis attack on the RSA variant with modulus N = prq for r ≥ 2 with three public and private exponents (e1, d1), (e2, d2), (e3, d3) sharing the same modulus N where p and q are consider to be primes having the same bit size. Our attack shows that we get the private exponent σ1σ2σ3 < ( r-1/r+1)4, which makes the modulus vulnerable to Coppersmith's attacks and can lead to the factorization of N efficiently where d1 < N..., d2 < N..., and d3 < N.... The asymptotic bound of our attack is greater than the bounds for May [1], Zheng and Hu [2], and Lu et al. [3] for 2 ≤ r ≤ 10 and greater than Sarkar's [4] and [5] bounds for 5 ≤ r ≤ 10. [ABSTRACT FROM AUTHOR]

Protecting ML classifiers from adversarial examples is crucial. We propose that the main threat is an attacker perturbing a confidently classified input to produce a confident misclassification. We consider in this paper the L 0 attack in which a small number of inputs can be perturbed by the attacker at test-time. To quantify the risk of this form of attack we have devised a formal guarantee in the form of an adversarial bound (AB) for a binary, Gaussian process classifier using the EQ kernel. This bound holds for the entire input domain, bounding the potential of any future adversarial attack to cause a confident misclassification. We explore how to extend to other kernels and investigate how to maximise the bound by altering the classifier (for example by using sparse approximations). We test the bound using a variety of datasets and show that it produces relevant and practical bounds for many of them. [ABSTRACT FROM AUTHOR]

We consider sequences of independent and identically absolutely continuously distributed random variables assuming that they have finite expectation and variance. We determine sharp lower and upper bounds on the expectation of the sum of n first sample maxima and n first upper record values under the condition that the value of the jth (1 ≤ j ≤ n) sample maximum and record value, respectively, are known and equal to a given quantile of the parent distribution. The bounds are expressed in terms of the expectation and standard deviation of the parent distribution. Analogous evaluations are presented for the sum of record values in n observations, when the jth sample maximum is known. The theoretical results are numerically compared.

Abstract The uncertainty relation is regarded as a remarkable feature of quantum mechanics differing from the classical counterpart, and it plays a backbone role in the region of quantum information theory. In principle, the uncertainty relation offers a nontrivial limit to predict the outcome of arbitrarily incompatible observed variables. Therefore, to pursue a more general uncertainty relations ought to be considerably important for obtaining accurate predictions of multi-observable measurement results in genuine multipartite systems. In this article, we derive a generalized entropic uncertainty relation (EUR) for multi-measurement in a multipartite framework. It is proved that the bound we proposed is stronger than the one derived from Renes et al. in [Phys. Rev. Lett. 103,020402(2009) ] for the arbitrary multipartite case. As an illustration, we take several typical scenarios that confirm that our proposed bound outperforms that presented by Renes et al. Hence, we believe our findings provide generalized uncertainty relations with regard to multi-measurement setting, and facilitate the EUR’s applications on quantum precision measurement regarding genuine multipartite systems.

Words are composed of morphemes. The notion that morphemes are words is not necessarily true. While some single morphemes can function as words, some words have two or more morphemes. Furthermore, it is wrong to consider morphemes to be syllables. In contrast, many words only have one syllable and two morphemes; examples are "cats," "runs," and "barked." Morpheme can be classified as either Free or Bound Morpheme. It applies the Bound Morpheme's restriction, which included eight inflectional affixes. The goal is to determine how pupils use bound morphemes. This study's research tool was acquired through testing. Knowing the sort of error and where it happens helps with testing. The procedures for data analysis were applied by the data analysis methodologies. Counted the Bound Morpheme first. Second, they were divided into groups. Thirdly, the number of frequently occurring terms was counted. Fourthly, determined which word should be chosen and which should not be. Then, based on the total number of words with the ratings Excellent, Good, Fair, and Poor, percentages were calculated. Meanwhile, based on the results of the The student's ability in implementing the Inflectional Affixes is 38% as a very good category, 25% as a good category, 25% as a fair category, and 12% as a low category in implementing the Inflectional Affixes of Bound Morpheme on the text. [ABSTRACT FROM AUTHOR]

The energy of a graph G, denoted by E (G) , is defined as the sum of the absolute values of all eigenvalues of G. In this paper, we present new lower bounds for the energy. Moreover, we improve some previous well-known bounds. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of relatively D -stable matrices and provide sufficient conditions for relative D -stability. We generalize the well-known Hadamard inequality, to provide upper bounds for the determinants of relatively D -stable and relatively additive D -stable matrices. For some classes of D -stable matrices, we estimate the sector gap between matrix spectra and the imaginary axis. We apply the developed technique to obtain upper bounds for determinants of some classes of D -stable matrices, e.g. diagonally stable, diagonally dominant and matrices with Q 2 -scalings. [ABSTRACT FROM AUTHOR]

The uncertainty relation is regarded as a remarkable feature of quantum mechanics differing from the classical counterpart, and it plays a backbone role in the region of quantum information theory. In principle, the uncertainty relation offers a nontrivial limit to predict the outcome of arbitrarily incompatible observed variables. Therefore, to pursue a more general uncertainty relations ought to be considerably important for obtaining accurate predictions of multi-observable measurement results in genuine multipartite systems. In this article, we derive a generalized entropic uncertainty relation (EUR) for multi-measurement in a multipartite framework. It is proved that the bound we proposed is stronger than the one derived from Renes et al. in [Phys. Rev. Lett. ] for the arbitrary multipartite case. As an illustration, we take several typical scenarios that confirm that our proposed bound outperforms that presented by Renes et al. Hence, we believe our findings provide generalized uncertainty relations with regard to multi-measurement setting, and facilitate the EUR’s applications on quantum precision measurement regarding genuine multipartite systems. [ABSTRACT FROM AUTHOR]

We define the wavy probability distributions on a subset and ∆-wavy probability distributions - two generalizations of the wavy probability distributions. A classification on the ∆-waviness is given. For the ∆-wavy probability distributions having normalization constant, we give a formula for this constant, to compute this constant. We show that the Potts model is a ∆-wavy probability distribution, where A is a partition which will be specified. For the normalization constant of Potts model, we give formulas and bounds. As to the formulas for this constant, we give two general formulas, one of them is simple while the other is more complicated, and based on independent sets, a formula for the Potts model on connected separable graphs - closed-form expressions are then obtained in several cases -, and a formula for the Potts model on graphs with a vertex of degree 2 - a recurrence relation is then obtained for the normalization constant of Potts model on Cn, the cycle graph with n vertices; the normalization constant of Ising model on Cn is computed using this relation. As to the bounds for the normalization constant, we present two ways to obtain such bounds; we illustrate these ways giving a general lower bound, and a lower bound and an upper one when the model is the Potts model on Gn,n, the square grid graph, n = 6k, k > 1 - two upper bounds for the free energy per site of this model are then obtained, one of them being in the limit. A sampling method for the ∆-wavy probability distributions is given and, as a result, a sampling method for the Potts is given. This method - that for the Potts model too - has two steps, Step 1 and Step 2, when |A| > 1 and one step, Step 2 only, when |A| = 1. For the Potts model, Step 1 is, in general, difficult. As to Step 2, for the Potts model too, using the Gibbs sampler in a generalized sense, we obtain an exact (not approximate) sampling method having p + 1 steps (p + 1 substeps of Step 2), where p = |I|, I is an independent set, better, a maximal independent set, best, a maximum independent set - for the Potts model on Gn1,n2,...,nd, the d-dimensional grid graph, d > 1, n1,n2,...,nd > 1, n1n2...nd > 2, we obtain an exact sampling method for half or half+1 vertices. [ABSTRACT FROM AUTHOR]

In this article, we study stochastic orders over an interval. Mainly, we focus on orders related to the Laplace transform. The results are then applied to obtain a bound for heavy-tailed distributions and are illustrated by some examples. We also indicate how these ordering relationships can be adapted to the classical risk model in order to derive a moment bound for ruin probability. Finally, we compare it with other existing bounds.

We prove that for any positive integer there exists a smallest positive integer such that for there exist no Agievich-primitive partitions of the space into affine subspaces of dimension . We give lower and upper bounds on the value and prove that . Results of the same type for partitions into coordinate subspaces are established. [ABSTRACT FROM AUTHOR]

The determination of bounds on the size of codes with given minimum distance is an important problem in the coding theory. In this paper, we construct codes based on partial linear maps of finite-dimensional vector spaces, define the measure of distance via rank function, and present several upper bounds and lower bounds on the size of these codes. [ABSTRACT FROM AUTHOR]

For a positive integer k, a graph G is said to have the property I k if each component of G has independence number at most k. The I 1 -admission number of a graph G, denoted by η (G , I 1) , is the cardinality of a smallest vertex subset D such that V (G) = N G [ D ] or each component of G - N G [ D ] is a clique. In this paper, we show that for a connected graph G of order n, if G ∉ { P 3 , C 6 } , then η (G , I 1) ≤ 2 n 7 , and the bound is sharp. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with n vertices. The transmission Tv of a vertex v is defined to be the sum of the distances from v to all other vertices in G, that is, Tv = Σu∈Vduv, where duv denotes the distance between u and v. Let T1,..., Tn be the transmission sequence of G. Let D = (d i j ) n × n be the distance matrix of G, and T be the transmission diagonal matrix diag(T1,...,Tn). The matrix Q (G) = T + D is called the distance signless Laplacian of G. In this paper, we provide the distance signless Laplacian spectrum of complete k-partite graph, and give some sharp lower and upper bounds on the distance signless Laplacian spectral radius q (G) . [ABSTRACT FROM AUTHOR]

The problem of finding bounds of time-averaged characteristics of dynamical systems, such as the bound on the mean energy dissipation rate in a turbulent flow governed by incompressible Navier–Stokes equations, is considered. It is shown that the direct method described by Seis (2015 J. Fluid Mech.777, 591–603. (doi:10.1017/jfm.2015.384)) and the auxiliary functional method by Chernyshenko et al. (2014 Phil. Trans. R. Soc. B372, 20130350. (doi:10.1098/rsta.2013.0350)) are related and can lead to the same bound. The well-known background flow method of Doering and Constantin is equivalent to the auxiliary functional method with a quadratic auxiliary functional. The known implementations of the direct method apparently also correspond to quadratic auxiliary functionals. The findings are illustrated by the analysis of the plane Couette flow. Three routes of further progress using non-quadratic auxiliary functionals and at the same time allowing to use the experience accumulated with the background flow method are proposed: making the balance parameter dependent on the energy, making the background flow time-dependent in a specific way and adding helicity to the auxiliary functional. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'. [ABSTRACT FROM AUTHOR]

In this study, we propose a methodology to obtain the approximate amplitude of a nonlinear forced system at resonance that satisfies Lyapunov's global stability criterion. This analytical method is formulated using the Lyapunov stability criterion and nonlinear approximations. The results obtained were compared with those obtained from the fourth-order Runge–Kutta method. The proposed methodology can approximate the amplitude of a nonlinear non-autonomous system to reasonably good accuracy. This theory will be useful in experimental vibration analysis to approximate the damping in a system. The proposed methodology can also be extended to other nonlinear forced systems. The results also showed that the vibration attenuation or stability of the system depends mainly on the linear coefficients and is least dependent on the cubic stiffness. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n and U a unicyclic graph with the same order. We firstly give a sharp bound for mG(μ), the multiplicity of a Laplacian eigenvalue μ of G. As a straightforward result, mU(1) ⩽ n − 2. We then provide two graph operations (i.e., grafting and shifting) on graph G for which the value of mG(1) is nondecreasing. As applications, we get the distribution of mU (1) for unicyclic graphs on n vertices. Moreover, for the two largest possible values of mU(1) ∈ {n − 5, n − 3}, the corresponding graphs U are completely determined. [ABSTRACT FROM AUTHOR]

Abstract In 2010, Huang and Liu introduced a useful parameter called the generalized μ-scrambling indices of a primitive digraph. In this paper, we give some bounds for μ-scrambling indices of some primitive digraphs with d loops and the digraphs attained the sharp upper bounds are provided.

The eccentric connectivity coindex has recently been introduced (Hua and Miao, 2019) as the total eccentricity sum of all pairs of non-adjacent vertices in a graph. Considering the total eccentricity product of non-adjacent vertex pairs, we introduce here another invariant of connected graphs called the second Zagreb eccentricity coindex. We study some mathematical properties of the eccentric connectivity coindex and second Zagreb eccentricity coindex. We also determine the extremal values of the second Zagreb eccentricity coindex over some specific families of graphs such as trees, unicyclic graphs, connected graphs, and connected bipartite graphs and describe the extremal graphs. Moreover, we compare the second Zagreb eccentricity coindex with the eccentric connectivity coindex and give directions for further studies.

In this paper, the authors provide several sharp upper and lower bounds for the Neuman–Sándor mean in terms of the arithmetic and contra-harmonic means, and present some new sharp inequalities involving hyperbolic sine function and hyperbolic cosine function. [ABSTRACT FROM AUTHOR]

We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. These lead to two new fundamental parameters of binary error-correcting codes, both of which measure the probability that the maximum likelihood decoder fails. We then derive various bounds for the cardinality and weight distribution of a binary code in terms of these new parameters, giving examples of codes meeting the bounds with equality. [ABSTRACT FROM AUTHOR]

In this paper, let A = (a i 1 i 2 ⋯ i m) ∈ R [ m , n ] A = (a i 1 i 2 ⋯ i m) ∈ R [ m , n ] , when m ≥ 4 m ≥ 4 , based on the condition | | x | | 2 = 1 | | x | | 2 = 1 , a new Z Z -eigenvalue localization set for tensors is given. And we extend the Geršhgorin-type localization set for Z-eigenvalues of fourth order tensors to higher order tensors. As an application, a sharper upper bound for the Z-spectral radius of nonnegative tensors is obtained. Let H H be a k k -uniform hypergraph with k ≥ 4 k ≥ 4 and A (H) A (H) be the adjacency tensor of H H , a new upper bound for the Z-spectral radius ρ (H) ρ (H) is also presented. Finally, a checkable sufficient condition for the positive definiteness of even-order tensors and asymptotically stability of time-invariant polynomial systems is also given. [ABSTRACT FROM AUTHOR]

The eccentric connectivity index is defined for a connected graph G as the summation of the terms ɛ G (u) + ɛ G (v) over all edges u v of G , in which ɛ G (u) denotes the eccentricity of the vertex u in G. In this paper, we study a multiplicative version of this graph invariant under the name multiplicative eccentric connectivity index. We present a sharp lower bound on the multiplicative eccentric connectivity index of n -vertex trees with fixed diameter and a sharp upper bound on the multiplicative eccentric connectivity index of n -vertex trees with given number of pendent vertices. Using these results, the trees on n vertices with the first three minimum and maximum values of the multiplicative eccentric connectivity index are characterized. Moreover, we present sharp bounds on the multiplicative eccentric connectivity index of connected graphs in terms of certain graph parameters including the order, size, radius, diameter, and number of universal vertices, and study the relationship between this invariant and several other vertex-eccentricity-based invariants. In particular, the minimal graphs with respect to the multiplicative eccentric connectivity index in the set of all connected graphs of given order, connected graphs of given size, and unicyclic and bicyclic graphs of given order are characterized and a Nordhaus–Gaddum-type result for this invariant is given. [ABSTRACT FROM AUTHOR]