Sum Of Eigenvalues Research Articles

Two-sided Lieb–Thirring bounds

We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb–Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of (−Δ + V + M)uM = 1 in Rd; here M∈R is chosen so that the operator is positive. We further prove that the infimum of (uM−1−M) is a lower bound for the ground state energy E0 and derive a simple iteration scheme converging to E0.

An algorithm and maximum deficiency energy of some graphs

The deficiency [Formula: see text] of a vertex [Formula: see text] is the deviation between the degree of the vertex [Formula: see text] and the maximum degree of the graph. The maximum deficiency matrix [Formula: see text] of a simple graph [Formula: see text] is square matrix whose [Formula: see text]th entry is max([Formula: see text], if the vertices [Formula: see text] and [Formula: see text] are adjacent and [Formula: see text], otherwise. Maximum deficiency energy [Formula: see text] is the absolute sum of eigenvalues of maximum deficiency matrix [Formula: see text]. In this paper, we find maximum deficiency energy of some classes of graphs. Moreover, we construct an algorithm and Python 3.8 program to find out spectrum and maximum deficiency energy of [Formula: see text].

Lieb–Thirring and Jensen sums for non-self-adjoint Schrödinger operators on the half-line

We prove upper and lower bounds for sums of eigenvalues of Lieb–Thirring type for non-self-adjoint Schrödinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be optimal in various senses by proving the lower bounds for specific potentials. We consider sums that correspond to both the critical and non-critical cases.

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.

Reversibility of linear cellular automata with intermediate boundary condition

<abstract><p>This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over $ \mathbb{Z}_2 $.</p></abstract>

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.

On Laplacian Eigenvalues of Wheel Graphs

Consider G to be a simple graph with n vertices and m edges, and L(G) to be a Laplacian matrix with Laplacian eigenvalues of μ1,μ2,…,μn=zero. Write Sk(G)=∑i=1kμi as the sum of the k-largest Laplacian eigenvalues of G, where k∈{1,2,…,n}. The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer’s conjecture states that Sk(G)≤m+k+12, where k=1,2,…,n. This paper proves Brouwer’s conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph Wn+1, which provides a better approximation for this upper bound using Brouwer’s conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of Wn+1, n≥3. We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves.

The sum of the k largest distance eigenvalues of graphs

Motivated by progresses on study of eigenvalue sum in adjacency matrix and Laplacian matrix, in this paper, we focus on the sum of the k largest distance eigenvalues of graphs. Denote by Sk(D(G))=λ1(D(G))+⋯+λk(D(G)) the sum of the k largest distance eigenvalues of G. We initially determine the extremal graph attaining the minimum Sk(D(G)) among all threshold graphs except for complete graphs. Besides, we prove that the complete r-partite graph attains the minimum Sk(D(G)) among all graphs with chromatic number χ(G)=r and we also give the lower bounds on Sk(D(G)) when G is Kr+1-free, which are extensions of previous results from Lin (2019) and Lu & Lu (2022), respectively.

On a magnetic Lieb–Thirring-type estimate and the stability of bipolarons in graphene

Two-dimensional Weyl–Dirac relativistic fermions have attracted tremendous interest in condensed matter as they mimic relativistic high-energy physics. This paper concerns two-dimensional Weyl–Dirac operators in the presence of magnetic fields, in addition to a short-range scalar electric potential of the Bessel–Macdonald-type, restricted to its positive spectral subspace. This operator emerges from the action of a pristine graphene-like QED3 model recently proposed by De Lima, Del Cima, and Miranda, “On the electron–polaron–electron–polaron scattering and Landau levels in pristine graphene-like quantum electrodynamics,” Eur. Phys. J. B93, 187 (2020). A magnetic Lieb–Thirring-type inequality à la Shen is derived for the sum of the negative eigenvalues of the magnetic Weyl–Dirac operators restricted to their positive spectral subspace. An application to the stability of bipolarons in graphene in the presence of magnetic fields is given.

Multi-kernel learning for multi-label classification with local Rademacher complexity

Multi-label classification aims to construct prediction models from input space to output space for multi-label datesets. However, the feature space of multi-label dataset and the hypothesis space of classifier are always complex. In order to obtain high performance multi-kernel learning algorithms, the core problem is to design algorithms that compress both hypothesis space and feature space. In this paper, by combining the local Rademacher complexity and Hilbert-Schmidt independence criterion (HSIC), an effective multi-kernel learning algorithm for multi-label classification is proposed to compress the feature space and hypothesis space simultaneously. Based on the tail sum of eigenvalues of the integral operator corresponding to kernels, the upper bound of local Rademacher complexity of linear functions class in the feature space is determined. The Hilbert-Schmidt independence criterion is applied to maximize the correlation between the convex combination of the input kernel matrices and the ideal kernel matrix in the label space. Finally, the Laplace regularization model is built by controlling the tail sum of eigenvalues of the integral operator corresponding to kernels of this criterion. Therefore, the above process obtains a combined kernel, simultaneously by sorting the coefficients of the combined kernel, a feature selection algorithm can be constructed. On the basis, the combined kernel can also be used to design a new binary classifier for multi-label classification. The experimental results verify that our proposed multi-kernel learning algorithms can effectively compress the hypothesis space, while the feature selection algorithms can effectively compress the feature space.

Bounding the sum of the largest signless Laplacian eigenvalues of a graph

We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.

A new extension of Chubanov’s method to symmetric cones

We propose a new variant of Chubanov’s method for solving the feasibility problem over the symmetric cone by extending Roos’s method (Optim Methods Softw 33(1):26–44, 2018) of solving the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound for the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (1) equivalent to that of Roos’s original method (2018) and superior to that of Lourenço et al.’s method (Math Program 173(1–2):117–149, 2019) when the symmetric cone is the nonnegative orthant, (2) superior to that of Lourenço et al.’s method (2019) when the symmetric cone is a Cartesian product of second-order cones, (3) equivalent to that of Lourenço et al.’s method (2019) when the symmetric cone is the simple positive semidefinite cone, and (4) superior to that of Pena and Soheili’s method (Math Program 166(1–2):87–111, 2017) for any simple symmetric cones under the feasibility assumption of the problem imposed in Pena and Soheili’s method (2017). We also conduct numerical experiments that compare the performance of our method with existing methods by generating strongly (but ill-conditioned) feasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.

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.

Analysis of the Model Characteristics in the North Atlantic Simulated by the NEMO Model with Data Assimilation

The main aim of this work is to study the spatial–temporal variability of the model’s physical and spectral characteristics in the process of assimilation of observed ocean surface height data from the AVISO (Archiving, Validating and Interpolation Satellite Observation) archive in combination with the NEMO (Nucleus for European Modeling of the Ocean) ocean circulation model for a period of two months. For data assimilation, the GKF (Generalized Kalman filter) method, previously developed by the authors, is used. The purpose of this work is to study the spatial–temporal structure of the simulated characteristics using decomposition into eigenvalues and eigenvectors (Karhunen–Loeve decomposition method). The feature of the GKF method is the fact that the constructed Kalman weight matrix multiplied by the vector of observational data can be represented as a weighted sum of eigenvectors and eigenvalues (spectral characteristics of the matrix), which describe the spatial and temporal structure of corrections to the model. The main investigations are focused on the North Atlantic. Their variability in time and space is estimated in this study. Calculations of the main ocean characteristics, such as the surface height, temperature, salinity, and the current velocities on the surface and in the depths, both with and without assimilation of observational data, over a time interval of 60 days, were performed by using a high-performance computing system. The calculation results have shown that the main spatial variability of characteristics after data assimilation is consistent with the localization of the currents in the North Atlantic.

Plant Traits Guide Species Selection in Vegetation Restoration for Soil and Water Conservation.

Great efforts have been made to improve the soil and water conservation capacity by restoring plant communities in different climatic and land-use types. However, how to select suitable species from local species pools that not only adapt to different site environments, but also achieve certain soil and water conservation capacities is a great challenge in vegetation restoration for practitioners and scientists. So far, little attention has been paid to plant functional response and effect traits related to environment resource and ecosystem functions. In this study, together with soil properties and ecohydrological functions, we measured the seven plant functional traits for the most common species in different restoration communities in a subtropical mountain ecosystem. Multivariate optimization analyses were performed to identify the functional effect types and functional response types based on specific plant traits. We found that the community-weighted means of traits differed significantly among the four community types, and the plant functional traits were strongly linked with soil physicochemical properties and ecohydrological functions. Based on three optimal effect traits (specific leaf area, leaf size, and specific root length) and two response traits (specific leaf area and leaf nitrogen concentration), seven functional effect types in relation to the soil and water conservation capacity (interception of canopy and stemflow, maximum water-holding capacity of litter, maximum water-holding capacity of soil, soil surface runoff, and soil erosion) and two plant functional response types to soil physicochemical properties were identified. The redundancy analysis showed that the sum of all canonical eigenvalues only accounted for 21.6% of the variation in functional response types, which suggests that community effects on soil and water conservation cannot explain the overall structure of community responses related to soil resources. The eight overlapping species between the plant functional response types and functional effect types were ultimately selected as the key species for vegetation restoration. Based on the above results, we offer an ecological basis for choosing the appropriate species based on functional traits, which may be very helpful for practitioners involved in ecological restoration and management.

Study of neutrino mass matrices with vanishing trace and one vanishing minor

In this work we carry out a systematic texture study of the neutrino mass matrix with the ansatzes - (i) one vanishing minor and (ii) the zero sum of the mass eigenvalues with the CP phases (henceforth vanishing trace). There are six possible textures of a neutrino mass matrix with one vanishing minor. The viability of each texture is checked with $3\sigma$ values of current neutrino data by drawing scatter plots. In our analysis we are motivated to use the ratio of solar to atmospheric mass-squared differences $R_{\nu}$ for its precise measurement (and also the atmospheric mixing angle $\theta_{23}$) to constrain phenomenologically first the Dirac CP phase $\delta$ in the range of $0^{\degree}-360^{\degree}$ for a given texture with the solutions of the constraint equations. Subsequently we employ this constrained $\delta$ to determine the range of completely unknown Majorana CP Phases ($\alpha$ and $\beta$) for all the viable textures. We also check the neutrinoless double beta decay rate, $|m_{ee}|$ and the Jarlskog invariant, $J_{cp}$ for the textures. Finally the symmetry realization of all the viable textures under the flavor symmetry group $Z_5$ via seesaw mechanism is implemented along with the FN mechanism to determine mass hierarchy structure.

Energy of Graph

By given the adjacency matrix, laplacian matrix of a graph we can find the set of eigenvalues of graph in order to discussed about the energy of graph and laplacian energy of graph. (i.e. the sum of eigenvalues of adjacency matrix and laplacian matrix of a graph is called the energy of graph) and the laplacian energy of graph is greater or equal to zero for any graph and is greater than zero for every connected graph with more or two vertices (i.e. the last eigenvalues of laplacian matrix is zero), according to several theorems about the energy of graph and the laplacian energy of graph that are described in this work; I discussed about energy of graph, laplacian energy of graph and comparing them here.

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.

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.