Absolute Eigenvalue Research Articles

Complex Fuzzy Dynamical Graphs and their Applications in Signals Processing

In this paper we introduce the concepts of complex fuzzy dynamic graphs, complex fuzzy diagonal matrices and complex fuzzy Laplacian matrices. We use these graphs and their laplacian matrices as mathematical framework for applications in Sciences, especially signals processing. We define absolute average eigenvalues of the Complex Laplacian matrices and explore the properties of these matrices with their eigenvalues. We develop an algorithm using the absolute eigenvalues of the Laplacian matrices and apply this algorithm to signal and systems. Our study begins by establishing the theoretical foundation of complex fuzzy dynamic graphs, highlighting their role to model within dynamic systems including two dimensional uncertainties. We investigates the complex fuzzy Laplacian matrices obtain from these graphs. Our main focus is on the absolute eigenvalues of these matrices, which hold a vital role into the graph’s structural characteristics and behavior. In the context of signals processing, the research demonstrates how these absolute eigenvalues serve as essential matrices for system characterization. This study presents novel methods to analyze signals on complex fuzzy dynamic graphs. These methods are particularly relevant in scenarios where signals are influenced by dynamic and uncertain environments.

Adjacency Sequence and Adjacency Spectrum of Power Fuzzy Graphs

Objectives: To find the adjacency sequence, spectrum of the power fuzzy graphs and discuss its properties. Methods: The spectrum and energy of the power fuzzy graphs are derived using the adjacency matrices. Energy of the power fuzzy graph is computed by adding the absolute eigenvalues of the adjacency matrix. Findings: The condition for a power fuzzy graph, 𝐺𝑓 𝑘 to be vertex regular in terms of adjacency sequence has been verified. The energy sequence for the power fuzzy graph with increasing 𝑘 has been established. Novelty: The concept of minimizing the interval of the edge membership values when the powers of fuzzy graph increase makes difference in the spectrum and energy of fuzzy graphs with the same initial structure. The concept of adjacency matrix, spectrum and energy of power fuzzy graph is applied in power supply load distribution system. Keywords: Fuzzy graph; Power fuzzy graph; Adjacency sequence; Spectrum; Energy

Computed Gyration Tensors of Knotted Chiral and Achiral Topological Stereoisomers of C60 Cyclocarbons.

The electronic origins of the computed optical rotations of the simplest chiral and achiral chemical knots with comparatively simple compositions and large, anticipated magnetoelectric polarizabilities are provided. Linear response theory (LRT) is used to calculate the gyration at 1064 nm of two knotted polyyne chains, topological stereoisomers of cyclo[60]carbon. One isomer is analogous to the trefoil knot with approximate D3 symmetry and the other to the figure eight knot with approximate S4 symmetry. The response in each case can be attributed largely to the magnetic dipole term that arises in a near degenerate E-like excited state. An oriented achiral figure eight knot is as optically active in some directions as the chiral knot in any direction, and its absolute eigenvalues are larger.

Estimating Posterior Sensitivities with Application to Structural Analysis of Bayesian Vector Autoregressions

The inherent feature of Bayesian empirical analysis is the dependence of posterior inference on prior parameters, which researchers typically specify. However, quantifying the magnitude of this dependence remains difficult. This article extends Infinitesimal Perturbation Analysis, widely used in classical simulation, to compute asymptotically unbiased and consistent sensitivities of posterior statistics with respect to prior parameters from Markov chain Monte Carlo inference via Gibbs sampling. The method demonstrates the possibility of efficiently computing the complete set of prior sensitivities for a wide range of posterior statistics, alongside the estimation algorithm using Automatic Differentiation. The method’s application is exemplified in Bayesian Vector Autoregression analysis of fiscal policy in U.S. macroeconomic time series data. The analysis assesses the sensitivities of posterior estimates, including the Impulse response functions and Forecast error variance decompositions, to prior parameters under common Minnesota shrinkage priors. The findings illuminate the significant and intricate influence of prior specification on the posterior distribution. This effect is particularly notable in crucial posterior statistics, such as the substantial absolute eigenvalue of the companion matrix, ultimately shaping the structural analysis.

Two-Point Localization Algorithm of a Magnetic Target Based on Tensor Geometric Invariant.

Currently, magnetic gradient tensor-based localization methods face challenges such as significant errors in geomagnetic field estimation, susceptibility to local optima in optimization algorithms, and inefficient performance. In addressing these issues, this article propose a two-point localization method under the constraint of overlaying geometric invariants. This method initially establishes the relationship between the target position and the magnetic gradient tensor by substituting an intermediate variable for the magnetic moment. Exploiting the property of the eigenvector corresponding to the minimum absolute eigenvalue being perpendicular to the target position vector, this constraint is superimposed to formulate a nonlinear system of equations of the target's position. In the process of determining the target position, the Nara method is employed for obtaining the initial values, followed by the utilization of the Levenberg-Marquardt algorithm to derive a precise solution. Experimental validation through both simulations and experiments confirms the effectiveness of the proposed method. The results demonstrate its capability to overcome the challenges faced by a single-point localization method in the presence of some errors in geomagnetic field estimation. In comparison to traditional two-point localization methods, the proposed method exhibits the highest precision. The localization outcomes under different noise conditions underscore the robust noise resistance and resilience of the proposed method.

Comparative analysis on the validity of golden ratio, tri-bimaximal, hexagonal and bimaximal neutrino mixing patterns under the radiative corrections

We study the comparative analysis on the validity of four popular leptonic mixing patterns viz, golden ratio (GR), tri-bimaximal (TBM), hexagonal mixing (HM) and bimaximal (BM), which are assumed to be defined at high energy scale, M R = 1016 GeV. We perform numerical analysis of low energy neutrino oscillation parameters through the renormalisation group equations (RGEs) in both standard model (SM) and minimal supersymmetric standard model (MSSM) with the inclusion of scale dependent vacuum expectation value (VEV). The sum of three absolute neutrino mass eigenvalues obtained from the latest Planck cosmological data, ∑∣m i ∣ < 0.12 eV, is used as a constraint in the analysis of these four leptonic mixing patterns. We consider both normal hierarchical (NH) model and inverted hierarchical (IH) model in the analysis of these four leptonic mixing patterns, with SUSY breaking scale at m s = 1 TeV and tanβ=67 . Among the four mixing patterns, GR is found to satisfy the latest Planck cosmological data, ∑∣m i ∣ < 0.12 eV. Further analysis on the effect of the variation of SUSY breaking scale 1 TeV ≤ m s ≤ 14 TeV is also studied for GR neutrino mixing pattern in NH.

Binding Number, $k$-Factor and Spectral Radius of Graphs

The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_{G}(X)|/|X|$ taken over all non-empty subsets $X$ of $V(G)$ such that $N_{G}(X)\neq V(G)$. The association between the binding number and toughness is intricately interconnected, as both metrics function as pivotal indicators for quantifying the vulnerability of a graph. The Brouwer-Gu Theorem asserts that for any $d$-regular connected graph $G$, the toughness $t(G)$ always at least $\frac{d}{\lambda}-1$, where $\lambda$ denotes the second largest absolute eigenvalue of the adjacency matrix. Inspired by the work of Brouwer and Gu, in this paper, we investigate $b(G)$ from spectral perspectives, and provide tight sufficient conditions in terms of the spectral radius of a graph $G$ to guarantee $b(G)\geq r$. The study of the existence of $k$-factors in graphs is a classic problem in graph theory. Katerinis and Woodall state that every graph with order $n\geq 4k-6$ satisfying $b(G)\geq 2$ contains a $k$-factor where $k\geq 2$. This leaves the following question: which $1$-binding graphs have a $k$-factor? In this paper, we also provide the spectral radius conditions of $1$-binding graphs to contain a perfect matching and a $2$-factor, respectively.

On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

The non-commuting graph, denoted by , is defined on a finite group , with its vertices are elements of excluding those in the center of . In this graph, two distinct vertices are adjacent whenever they do not commute in . The graph can be associated with several matrices including the most basic matrix, which is the adjacency matrix, , and a matrix called Sombor matrix, denoted by . The entries of are either the square root of the sum of the squares of degrees of two distinct adjacent vertices, or zero otherwise. Consequently, the adjacency and Sombor energies of is the sum of the absolute eigenvalues of the adjacency and Sombor matrices of , respectively, whereas the spectral radius of is the maximum absolute eigenvalues. Throughout this paper, we find the spectral radius obtained from the spectrum of and the Sombor energy of for dihedral groups of order , where . Moreover, there is an almost linear correlation between the Sombor energy and the adjacency energy of for which is slightly different than reported earlier in previous literature.

Modified Sombor Spectral Radii and Modified Sombor Energies of Splitting and Shadow Graphs

Gutman et al. introduced Sombor and Modified Somber index because of the rapidly growing applications in chemistry and network analysis. Graph energy ε(G) and the spectral radius ℘(G) of graph G are essential components that are associated with the eigenvalues of the matrix of graph G and, chemically, with the intermolecular forces. These graph invariants have many useful applications in computer sciences, networking, and molecular computing. There are numerous variants of the ε(G) and ℘(G) attained by substituting another matrix in place of an adjacency matrix. Modified Sombor energy, MSε(G) is defined as the sum of absolute eigenvalues of the modified Sombor matrix or in other words we can say that modified Sombor energy, MSε(G) is the trace norm of modified Sombor matrix. Modified Sombor spectral radius, ℘MS is defined as the largest absolute eigenvalue of the modified Sombor matrix. The major focus of this article is on the MSε(G), ℘MS of the generalized shadow and splitting graphs. The only realistic problem in which we are particularly interested in how MSε(Splt(G)) and MSε(Sht(G)) are comparable to MSε(G). On similar lines we are also interested in how ℘MS(Splt(G)) and ℘MS(Sht(G)) are comparable to ℘MS(G). We were able to address these challenges by focusing on splitting and shadow graphs.

Wiener index and graph energy of zero divisor graph for commutative rings

Let [Formula: see text] be commutative ring and [Formula: see text] be the set of all non-zero zero divisors of [Formula: see text]. Then [Formula: see text] is said to be the zero divisor graph if and only if [Formula: see text] where [Formula: see text] and [Formula: see text]. Graph energy [Formula: see text] is defined as the sum of the absolute eigenvalues of the adjacency matrix [Formula: see text], then [Formula: see text]. Wiener index [Formula: see text] is defined as the sum of all distance between pairs of vertices [Formula: see text] and [Formula: see text], then [Formula: see text]. In this paper, we compute the graph energy from the adjacency matrix of the zero divisor graph and the Wiener index from the zero divisor graph associated with commutative rings.

Albertson (Alb) spectral radii and Albertson (Alb) energies of graph operation.

The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.

ISI spectral radii and ISI energies of graph operations

Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph’s spectral radius represents the adjacency matrix’s largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article’s main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph.

Communication area estimation for decentralized control of nanosatellites swarm

The problem of relative drift elimination between the satellites in the swarm is considered in the paper. The proposed decentralized control takes into account a communication constraint such as limited size of communication area. Only the satellites within the communication area can be identified by relative motion determination system. The control aim is to eliminate the mean relative drift between all the satellites inside the communication area. The purpose of the work is to study the performance of the proposed decentralized control algorithm. It is shown that the system matrix of differential equations for the vector of relative drifts is related to the Laplacian matrix of the communication graph. In the case of the connected swarm, all but one eigenvalues of the system are negative, and the remaining one is equal to zero. It means that all the relative drifts converge to the same value under the proposed control. The speed of convergence is defined by the minimum absolute eigenvalue that depends on the graph topology. The initial drift and the convergence speed make it possible to estimate the communication distance that provides the connectivity of the graph. Considering normally distributed errors of the initial velocity after the launch, it is possible to estimate the distance between any two satellites after the convergence. It allows us to estimate the communication distance that ensures the relative drift elimination between all the satellites in the swarm. The obtained estimations are validated using Monte Carlo simulations. In numerical simulations the swarm of 3U CubeSats in low-Earth orbit is considered. The decentralized control is implemented by differential aerodynamic drag via the change of cross-sectional area using onboard reaction wheels.

Vertical detachment energies of ammonia cluster anions using self-interaction-corrected methods.

Systems with weakly bound extra electrons impose great challenges to semilocal density functional approximations (DFAs), which suffer from self-interaction errors. Small ammonia clusters are one such example of weakly bound anions where the extra electron is weakly bound. We applied two self-interaction correction (SIC) schemes, viz., the well-known Perdew-Zunger and the recently developed locally scaled SIC (LSIC) with the local spin density approximation (LSDA), Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and the SCAN meta-GGA functionals to calculate the vertical detachment energies (VDEs) of small ammonia cluster anions (NH3)n-. Our results show that the LSIC significantly reduces the errors in calculations of VDE with LSDA and PBE-GGA functionals leading to better agreement with the reference values calculated with coupled cluster singles and doubles with perturbative triples [CCSD(T)]. Accurate prediction of VDE as an absolute of the highest occupied molecular orbital (HOMO) is challenging for DFAs. Our results show that VDEs estimated from the negative of HOMO eigenvalues with the LSIC-LSDA and Perdew-Zunger SIC-PBE are within 11meV of the reference CCSD(T) results. The LSIC method performs consistently well for the VDE estimates, from both the total energy differences and the absolute HOMO eigenvalues.

Randic and reciprocal randic spectral radii and energies of some graph operations

The largest absolute eigenvalue of a matrix A associated to the graph G is called the A-Spectral Radius of the graph G, and A-energy of the graph G is defined as the absolute sum of all its eigenvalues. In the present article, we compute Randic energies, Reciprocal Randic energies, Randic spectral radii and Reciprocal Randic radii of s-shadow and s-splitting graph of G. We actually relate these energies and Spectral Radii of new graphs with the energies and Spectral Radii of original graphs.

Neighbors Degree Sum Energy of Commuting and Non-Commuting Graphs for Dihedral Groups

The neighbors degree sum (NDS) energy of a graph is determined by the sum of its absolute eigenvalues from its corresponding neighbors degree sum matrix. The non-diagonal entries of NDS−matrix are the summation of the degree of two adjacent vertices, or it is zero for non-adjacent vertices, whereas for the diagonal entries are the negative of the square of vertex degree. This study presents the formulas of neighbors degree sum energies of commuting and non-commuting graphs for dihedral groups of order 2n, D2n, for two cases−odd and even n. The results in this paper comply with the well known fact that energy of a graph is neither an odd integer nor a square root of an odd integer.

Toughness, hamiltonicity and spectral radius in graphs

The study of the existence of Hamiltonian cycles in a graph is a classical problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chvátal’s conjecture from another perspective: What is the spectral condition to guarantee the existence of a Hamiltonian cycle among t-tough graphs? We first give the answer to 1-tough graphs, i.e. if ρ(G)≥ρ(Mn), then G contains a Hamiltonian cycle, unless G≅Mn, where Mn=K1∇Kn−4+3 and Kn−4+3 is the graph obtained from 3K1∪Kn−4 by adding three independent edges between 3K1 and Kn−4. The Brouwer’s toughness theorem states that every d-regular connected graph always has t(G)>dλ−1 where λ is the second largest absolute eigenvalue of the adjacency matrix. In this paper, we extend the result in terms of its spectral radius, i.e. we provide a spectral condition for a graph to be 1-tough with minimum degree δ and to be t-tough, respectively.

A Fourier-transformed feature engineering design for predicting ternary perovskite properties by coupling a two-dimensional convolutional neural network with a support vector machine (Conv2D-SVM)

In computational material sciences, Machine Learning (ML) techniques are now competitive alternatives that can be used in determining target properties conventionally resolved by ab initio quantum mechanical simulations or experimental synthesization. The successes realized with ML-based techniques often rely on the quality of the design architecture, in addition to the descriptors used in representing a chemical compound with good target mapping property. With the perovskite crystal structure at the forefront of modern energy materials discovery, accurately estimating related target properties is even of high importance due to the role such properties may have in defining the functionalization. As a result, the present study proposes a new feature engineering approach that takes advantage of both the direct ionic features and the periodic Fourier transformed reciprocal features of a three-dimensional perovskite polyhedral. The study is conducted on about 27,000 ABX3 perovskite structures with the stability energy, the formation energy, and the energy bandgap as targets. For accurate modeling, a feature-extracting two-dimensional convolutional neural network (Conv2D) is coupled with a prediction-enhancing Support Vector Machine (SVM) to form a hybridized Conv2D-SVM architecture. A comparison with previous benchmark evaluations reveals appreciable improvements in modeling accuracy for all target properties, particularly for the energy bandgap, for which the feature extraction approach yields 0.105 eV MAE, 0.301 eV RMSE, and 93.48% R2. Besides, the proposed design is further demonstrated to out-perform other similar periodic feature engineering approaches in the Coulomb matrix, Ewald-sum matrix, and Sine matrix, all in their absolute eigenvalue forms. All preprocessed data, source codes, and relevant sample calculations are openly available at: github.com/chenebuah/high_dim_descriptor.

Maximum and Minimum Degree Energy of Commuting Graph for Dihedral Groups

If G is a finite group and Z(G) is the centre of G, then the commuting graph for G, denoted by ΓG, has G\Z(G) as its vertices set with two distinct vertices vp and vq are adjacent if vp vq = vq vp. The degree of the vertex vp of ΓG, denoted by 𝑑𝑑𝑣𝑣𝑝𝑝 , is the number of vertices adjacent to vp. The maximum (or minimum) degree matrix of ΓG is a square matrix whose (p,q)-th entry is max{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 } (or min{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 }) whenever vp and vq are adjacent, otherwise, it is zero. This study presents the maximum and minimum degree energies of ΓG for dihedral groups of order 2n, D2n by using the absolute eigenvalues of the corresponding maximum degree matrices (MaxD(ΓG)) and minimum degree matrices (MinD(ΓG)). Here, the comparison of maximum and minimum degree energy of ΓG for D2n is discussed by considering odd and even n cases. The result shows that for each case, both energies are non-negative even integers and always equal.

On Some Topological Indices Defined via the Modified Sombor Matrix.

Let G be a simple graph with the vertex set and denote by the degree of the vertex . The modified Sombor index of G is the addition of the numbers over all of the edges of G. The modified Sombor matrix of G is the n by n matrix such that its -entry is equal to when and are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of . The sum of the absolute eigenvalues of is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is ; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven.