Laplacian Energy Research Articles

On the Laplacian Energy of an Orbit Graph of Finite Groups

Let β H denote the orbit graph of a finite group H . Let ζ be the set of commuting elements in H with order two. An orbit graph is a simple undirected graph where non-central orbits are represented as vertices in ζ , and two vertices in ζ are connected by an edge if they are conjugate. In this article, we explore the Laplacian energy and signless Laplacian energy of orbit graphs associated with dihedral groups of order $2w$ and quaternion groups of order 2 w .

Hermitian skew Laplacian matrix of oriented graphs

Let [Formula: see text] be a simple graph of order [Formula: see text] having [Formula: see text] edges and let [Formula: see text] be an orientation of [Formula: see text]. The hermitian skew Laplacian matrix [Formula: see text] of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the imaginary unit, [Formula: see text] is the diagonal matrix of oriented degrees [Formula: see text] and [Formula: see text] is the skew matrix. The spectral radius of the matrix [Formula: see text] is the hermitian skew Laplacian spectral radius and the sum of the absolute values of eigenvalues of [Formula: see text] is the hermitian skew Laplacian energy of [Formula: see text]. In this paper we study the hermitian spectral radius and hermitian skew Laplacian energy of [Formula: see text]. We obtain some new bounds for the hermitian skew Laplacian spectral radius and characterize the extremal oriented graphs attaining these bounds. Further, we obtain some bounds for the hermitian skew Laplacian energy and locate the extremal oriented graphs.

Predictive potential of eigenvalues-based graphical indices for determining thermodynamic properties of polycyclic aromatic hydrocarbons with applications to polyacenes

For the structure–property modeling of thermodynamic properties of polycyclic aromatic hydrocarbons (PAHs), this paper explores the effectiveness of a specific class of graphical indices defined based on eigenvalues, also known as valency-spectral graphical indices. The 30 lower PAHs have been selected as test molecules. The heat capacity (Cp) and entropy (So) are the standard choices for characterizing thermodynamic properties due to their close correlation with the thermodynamic properties of a similar nature. A computer-dependent method is proposed to calculate valency-spectral indices of chemical graphs, and it is employed to compute the indices for our test molecules. The statistical inferences reveal an unexpected finding for the Estrada harmonic energy, a less familiarized spectral descriptor within the research community, outperforming all the existing descriptors, followed by the positive & negative inertia indices. The Estrada harmonic energy achieved the highest mean correlation coefficient of ρ>0.99. On the other hand, well-studied descriptors such as Laplacian energy and adjacency energy still maintained their effectiveness in correlating the thermodynamic properties of PAHs; however, they were significantly outperformed by the Estrada harmonic energy. The three best valency-spectral indices are employed in the structure–property modeling of thermodynamic properties of linear polyacenes.

Seidel Laplacian Energy of Fuzzy graphs

The energy of a graph is related to its spectrum, which is equal to the total of the latent values of the pertinent adjacency matrix. In this research work, we proposed some of the features and the energy of the Seidel Laplacian of a fuzzy graph. Also, the lower and upper bounds for the energy of the Seidel Laplacian of a fuzzy graph were studied with suitable illustrative examples.

The Laplacian spectrum of weighted composite networks and the applications

The topological properties of the networks can be described by the Laplacian spectra, but resolving the Laplacian spectra of networks poses difficulties. In this study, a novel approach for solving the Laplacian spectrum of weighted composite networks is presented. We first give the definitions of three weighted graph operations, namely, Cartesian product, corona, and join. Second, the Laplacian spectra of these composite networks are calculated. Finally, we use the obtained Laplacian spectrum to deduce some topological properties of the networks, such as network coherence, entire mean first-passage time, and Laplacian energy, which have several applications in physical chemistry.

Identifying critical nodes in complex networks based on distance Laplacian energy

Identifying critical nodes in complex networks is a fundamental problem, it plays a crucial role in stabilizing the performance of the network structure and propagating information. Majority of the existing studies are built by directly considering the topology of the network. In this paper, a new vertex centrality called distance Laplacian centrality (DLC) is proposed for critical nodes identification from the perspective of graph energy. This method incorporates the vertex’s transfer degree, considers the position of nodes in the network from a global perspective, and measures the importance of a node using the relative variation of the distance Laplacian energy responding to the deletion of the node from the network. To validate the performance and applicability of the proposed method, this paper compares DLC with other methods through susceptible-infected-recovered (SIR) model on different real networks. The experimental results demonstrate that DLC has better performance in terms of influence, distinguishing ability relevance and ranking accuracy, and can effectively recognize critical nodes in complex networks.

The Energy of Interval-Valued Complex Neutrosophic Graph Structures: Framework, Application and Future Research Directions


 Graph structure is a developing field with many real-world applications and advancements, particularly effective frameworks for integrative problem-solving in computer networks and artificial intelligence systems. To define the idea of an Interval-Valued Complex Neutrosophic Graph Structure (IVCNGS), the concept of an Interval-Valued Complex Neutrosophic Set (IVCNS) is applied to the graph structure. Using the adjacency matrix to calculate the degree of vertex, we have defined some findings about the IVCNGS. Further, we compute the energy and Laplacian energy of IVCNGS. Moreover, we derive the lower and upper bounds for the energy and Laplacian energy of IVCNGS, and we have discussed their application in IVCNGS. Finally, we develop an algorithm that clarifies the fundamental processes of the application.

An assessment of fertilizer spraying drones based on hesitancy fuzzy similarity measures for sustainable green development

This research proposes a new similarity measure on hesitancy fuzzy graphs. The similarity measures are crucial concepts to explore the closeness between fuzzy graphs. The available imprecise and inconsistent data were more effectively handled by fuzzy similarity measures and intuitionistic fuzzy similarity measures. With time in decision-making theory, a complex frame of the background that occurs cannot be specified entirely by these fuzzy graphs but generalized fuzzy graphs like the hesitancy fuzzy graph can handle such a situation efficiently. The applicability of Hesitancy Fuzzy Graph (HFG) attracted the researchers to generalize ranking order based on the working procedures I, II of Xu’s approach and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) approach. For this purpose, we first define hesitancy fuzzy preference relations, Laplacian energy, and similarity measures in these HFG. The decision-making technique is then proposed in which a weighting technique is developed by building an optimal model based on the proposed Laplacian energy and similarity measure under a hesitancy fuzzy environment. This research studies the decision-making problem in which the preference information given by the experts takes the form of Hesitancy Fuzzy Preference Relations (HFPRs) and the information about experts’ weights are completely unknown. This research utilizes the hesitancy fuzzy weighted averaging operator to aggregate all individual HFPRs into a collective HFPR. Then, based on the degree of similarity between the individual HFPRs and the collective ones, we develop an approach to determine the experts’ weights. Moreover, based on HFPRs, in which the similarity measures between the collective preference relation and hesitancy fuzzy ideal solution are used to rank the given alternatives. We provide an illustrative numerical example to illustrate the mentioned approach and also we compare the aggregate results of the two techniques.

On the Eigenvalues and Energy of the Seidel and Seidel Laplacian Matrices of Graphs

Let S(Γ) be a Seidel matrix of a graph Γ of order n and let D(Γ) = diag(n − 1 − 2d1, n − 1 − 2d2, …, n − 1 − 2dn) be a diagonal matrix with di denoting the degree of a vertex vi in Γ. The Seidel Laplacian matrix of Γ is defined as SL(Γ) = D(Γ) − S(Γ). In this paper, we obtain an upper bound, and a lower bound on the Seidel Laplacian Estrada index of graphs. Moreover, we find a relation between Seidel energy and Seidel Laplacian energy of graphs. We establish some lower bounds on the Seidel Laplacian energy in terms of different graph parameters. Finally, we present a relation between Seidel Laplacian Estrada index and Seidel Laplacian energy of graphs.

A study on centrality measures in weighted networks: A case of the aviation network

<abstract><p>Identifying influential spreaders in complex networks is a crucial issue that can help control the propagation process in complex networks. An aviation network is a typical complex network, and accurately identifying the key city nodes in the aviation network can help us better prevent network attacks and control the spread of diseases. In this paper, a method for identifying key nodes in undirected weighted networks, called weighted Laplacian energy centrality, was proposed and applied to an aviation network constructed from real flight data. Based on the analysis of the topological structure of the network, the paper recognized critical cities in this network, then simulation experiments were conducted on key city nodes from the perspectives of network dynamics and robustness. The results indicated that, compared with other methods, weighted Laplacian energy centrality can identify the city nodes with the most spreading influence in the network. From the perspective of network robustness, the identified key nodes also have the characteristics of accurately and quickly destroying network robustness.</p></abstract>

Reciprocal distance Laplacian spectral properties double stars and their complements

Several matrices are associated with graphs in order to study their properties. In such a study, researchers are interested in the spectra of the matrix under consideration, therefore, the properties are called spectral properties, with reference to the matrix. One of the interesting and hard problems in the spectral study of graphs is to order the graphs based on some spectral graph invariant, like the spectral radius, the second smallest eigenvalue, the energy, etc. Due to hardness of this problem it has been considered in the literature for small classes of graphs. Here we continue this study and add some more classes of graphs which can be ordered on the basis of spectral graph invariants. In this article, we study spectral properties of trees of diameter three, called double stars, and their complements through their reciprocal distance Laplacian eigenvalues. We give ordering of these graphs based on their reciprocal distance Laplacian spectral radius, on their second smallest reciprocal distance Laplacian eigenvalue, and on their reciprocal distance Laplacian energy.

Inequalities involving energy and Laplacian energy of non-commuting graphs of finite groups

Inequalities involving energy and Laplacian energy of non-commuting graphs of finite groups

Neural-Singular-Hessian: Implicit Neural Representation of Unoriented Point Clouds by Enforcing Singular Hessian

Neural implicit representation is a promising approach for reconstructing surfaces from point clouds. Existing methods combine various regularization terms, such as the Eikonal and Laplacian energy terms, to enforce the learned neural function to possess the properties of a Signed Distance Function (SDF). However, inferring the actual topology and geometry of the underlying surface from poor-quality unoriented point clouds remains challenging. In accordance with Differential Geometry, the Hessian of the SDF is singular for points within the differential thin-shell space surrounding the surface. Our approach enforces the Hessian of the neural implicit function to have a zero determinant for points near the surface. This technique aligns the gradients for a near-surface point and its on-surface projection point, producing a rough but faithful shape within just a few iterations. By annealing the weight of the singular-Hessian term, our approach ultimately produces a high-fidelity reconstruction result. Extensive experimental results demonstrate that our approach effectively suppresses ghost geometry and recovers details from unoriented point clouds with better expressiveness than existing fitting-based methods.

From Ultimate Energy to Graph Energies

A lower bound is given for the ultimate energy, which is applicable to all graph energies. The extremal characterization is provided for the energy and Laplacian energy when the graph is connected.

On the signless Laplacian energy of a digraph

On the signless Laplacian energy of a digraph

HyperSLICE: HyperBand optimized spiral for low-latency interactive cardiac examination.

Interactive cardiac MRI is used for fast scan planning and MR-guided interventions. However, the requirement for real-time acquisition and near-real-time visualization constrains the achievable spatio-temporal resolution. This study aims to improve interactive imaging resolution through optimization of undersampled spiral sampling and leveraging of deep learning for low-latency reconstruction (deep artifact suppression). A variable density spiral trajectory was parametrized and optimized via HyperBand to provide the best candidate trajectory for rapid deep artifact suppression. Training data consisted of 692 breath-held CINEs. The developed interactive sequence was tested in simulations and prospectively in 13 subjects (10 for image evaluation, 2 during catheterization, 1 during exercise). In the prospective study, the optimized framework-HyperSLICE- was compared with conventional Cartesian real-time and breath-hold CINE imaging in terms quantitative and qualitative image metrics. Statistical differences were tested using Friedman chi-squared tests with post hoc Nemenyi test (p < 0.05). In simulations the normalized RMS error, peak SNR, structural similarity, and Laplacian energy were all statistically significantly higher using optimized spiral compared to radial and uniform spiral sampling, particularly after scan plan changes (structural similarity: 0.71 vs. 0.45 and 0.43). Prospectively, HyperSLICE enabled a higher spatial and temporal resolution than conventional Cartesian real-time imaging. The pipeline was demonstrated in patients during catheter pull back, showing sufficiently fast reconstruction for interactive imaging. HyperSLICE enables high spatial and temporal resolution interactive imaging. Optimizing the spiral sampling enabled better overall image quality and superior handling of image transitions compared with radial and uniform spiral trajectories.

Bounds on the parameters of non- L -borderenergetic graphs

UDC 519.17 We consider graphs such that their Laplacian energy is equivalent to the Laplacian energy of the complete graph of the same order, which is called an L -borderenergetic graph. Firstly, we study the graphs with degree sequence consisting of at most three distinct integers and give new bounds for the number of vertices of these graphs to be non- L -borderenergetic. Second, by using Koolen–Moulton and McClelland inequalities, we give new bounds for the number of edges of a non- L -borderenergetic graph. Third, we use recent bounds given by Milovanovic, et al. on Laplacian energy to get similar conditions for non- L -borderenergetic graphs. Our bounds depend only on the degree sequence of a graph, which is much easier than computing the spectrum of the graph. In other words, we developed a faster approach to exclude non- L -borderenergetic graphs.

Applications on color (distance) signless laplacian energy of annihilator monic prime graph of commutative rings

In this study, we define the structure formation of the annihilator monic prime graph of commutative rings, whose distinct vertices X and J satisfies a condition annXJ≠annX⋃ann(J), graph is denoted by AMPG(Zn[x]/〈fx〉). The chromatic number of the annihilator monic prime graph is determined to establish the color (signless laplacian) energy of the graph. Also, we introduce color based energy of the annihilator monic prime graph, namely color distance signless Laplacian energy. Diag(Tr)+D(G)+Aχ, where Diag(Tr) denotes the diagonal matrix of the vertex transmissions in the graph, D(G) is the distance matrix of vertices whose diagonals are zero, the entries of the matrix Aχ are 1 if ηi and ηj are adjacent, −1 if ηi and ηj are non-adjacent with different colors, otherwise 0. The eigenvalues of color distance signless Laplacian matrix will be denoted by ∂1χ+≤∂2χ+≤∂3χ+≤⋯≤∂nχ+, then DSLχ+ of annihilator monic prime graph are define as EDSLχ+Znx〈fx〉=∑1n′∂iχ+-tG where t(G) is the transmission of the graph. Based on the eigenvalues, we can study some more properties of graphs. Applications of color based energy are also discuss in this paper.

Energy based bipolar intuitionistic fuzzy digraph decision-making system in selecting COVID-19 vaccines

The energy of a graph, defined as the absolute sum of the eigenvalues (EIGs) of its adjacency matrix, has been applied to numerous research fields, including decision-making techniques. It can be defined for a fuzzy graph when the graph has uncertain vertices and edges. A bipolar intuitionistic fuzzy graph (BIFG) is an extended version of fuzzy graph that helps to deal with uncertainty in an effective manner. Therefore, this study unites the above-said concepts to examine the energy concepts with additional information and utilizes them in the decision-making problem to explore their applicability. Initially, the notions, relations and bounds of the energy, Laplacian energy (LE) and signless Laplacian energy (SLE) are defined. Further, a novel energy-based multi-criteria decision-making (MCDM) technique is designed by utilizing the energy of BIFG to obtain the objective weights for the decision experts. The selection problem of five COVID-19 vaccines with respect to eight important criteria is taken to elucidate the proposed decision-making technique, where Moderna has resulted as the finest vaccine among other vaccines. Moreover, a comparative is done with the correlation and distance measures to show the superiority of the proposed method. Finally, a sensitivity analysis is done to depict the reliability of the proposed technique.

ENERGY OF FIBONACCI PRODUCT CORDIAL GRAPH

Let G be a Fibonacci product cordial graph with n vertices. Then the Fibonacci product cordial label energy of G is denoted by EFPCL [G] and is defined as EFPCL[G]= n i=1 |ρi|, where ρi are the eigenvalues of the Fibonacci product cordial labeled matrix. In this paper, we introduce the Fibonacci product cordial labeled matrix, Fibonacci Product Cordial Labeled Laplacian matrix, Fibonacci product cordial label Laplacian energy, Fibonacci product cordial label equi-laplacian energic graphs respectively.