Energy Of Graph Research Articles

Optimizing Warehouse Selection Through Topological Index Based Energy of q-Rung Orthopair Fuzzy Graphs

In recent years, the study of q-rung orthopair fuzzy graphs (or q-ROFGs) has gained significant attention due to their ability to model complex and uncertain relationships. One crucial aspect in analyzing such graphs is the computation of their energy, which provides insights into their structural properties and dynamics. q-rung orthopair fuzzy sets were introduced, as a method for adjusting the range of indication of decision information by modifying a parameter q. This article presents a comprehensive approach for computing the energy of q-ROFGs. The new concepts of Zagreb energy of q-ROFGs are proposed and some bounds are determined. To demonstrate the applicability of the developed approach and show its viability, a numerical model related to the selection of the most suitable warehouse location is provided. Lastly, a comparative analysis is done, and the proposed approach is found to be compatible with existing models.

Energy of Fuzzy graphs

Energy of Fuzzy graphs

Bounds for the Energy of Hypergraphs

The concept of the energy of a graph has been widely explored in the field of mathematical chemistry and is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of a hypergraph is the trace norm of its connectivity matrices, which generalize the concept of graph energy. In this paper, we establish bounds for the adjacency energy of hypergraphs in terms of the number of vertices, maximum degree, eigenvalues, and the norm of the adjacency matrix. Additionally, we compute the sum of squares of adjacency eigenvalues of linear k-hypergraphs and derive its bounds for k-hypergraph in terms of number of vertices and uniformity of the k-hypergraph. Moreover, we determine the Nordhaus–Gaddum type bounds for the adjacency energy of k-hypergraphs.

Enhancing Expert Decision-Making for Wastewater Treatment Plants with Seidel Laplacian Energy and Cosine Similarity Measure in Intuitionistic Fuzzy Graphs

Wastewater treatment facilities’ main goal is to protect the public and environment from the hazardous and poisonous materials found in wastewater. Water treatment facilities were developed to speed up the natural process of cleansing water. A novel cosine similarity measure across intuitionistic fuzzy graphs has been proven to be more effective than certain present ones in group decision-making issues using example verification. This paper provides a unique approach for calculating expert-certified, well-known scores by finding the ambiguous information of intuitionistic fuzzy preference relations as well as the regular cosine similarity grades from one separable intuitionistic fuzzy preference relation to another. The new technique considers both "objective" and "subjective" information provided by experts. Using intuitionistic fuzzy preference relations, we provide workable techniques for judging experts’ eligible reputational ratings. This can be used to raise or decrease the relevance of the stated criteria in an evaluation that takes into account several competing elements. We give a solution to a decisional problem by using two effective methods: the newly constructed cosine similarity measure and the Seidel Laplacian energy (SLe+) of an intuitionistic fuzzy graph. Finally, two working procedures and circumstances are offered to show the effectiveness and superiority of the proposed techniques.

Szeged and Padmakar-Ivan Energies of Non-Commuting Graph for Dihedral Groups

The graph can represent the molecule structure and the -electron energy derives the concept of graph energy. The graph also can be related to the groups or rings as its vertex set. The non-commutative graph is a type of graph whose construction is determined by the structure of a group. This paper focuses on the energy of the non-commuting graph for dihedral groups using the Szeged and Padmakar-Ivan matrices. Both matrices are constructed based on the distance between two vertices in the graph. The eigenvalues ​​of these matrices lead to the formulation of the graph's energy values​. Interestingly, the energies obtained are equal to twice the spectral radius and are hyperenergetic.

Distance based topological characterization, graph energy prediction, and NMR patterns of benzene ring embedded in P-type surface in 2D network

Nanostructures are tiny objects at the molecular and microscopic scale, with carbon nanotubes being the most notable among them. The elements possess exceptional microelectronic properties and other unique characteristics. Researchers have recently focused on the mathematical features of these materials. Molecular descriptors are crucial in mathematical chemistry, particularly in QSAR and QSPR modeling. Topological indices hold a significant position among them. This study presents the precise formulation of the ten most crucial topological indices for a benzene ring positioned on a P-type surface within the highly symmetric 2D lattice BCZ48\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {BCZ}_{48}$$\\end{document}. We have incorporated the computed indices to develop a predictive model for the graph energy of the 2D lattice and, in addition, provided the NMR patterns and the HOMO-LUMO gap.

Partition energy of m-splitting graph and their generalized complements

Let G be graph with partition P={V1, V2, …, Vk} of the vertex set V. Recently E. Sampathkumar et al. [22] introduced the concept of k-partition energy of graph EPₖ (G) and studied partition energy of certain class of graphs. In this paper we compute partition energy of some m-splitting graphs Sm(G). The graphs we consider are Sm (Kn), Sm(Cn), Sm(Kn×2), Sm(Kn,n) and their generalized complements.

On the Various Energy Forms of the Join of Complete Graphs

The concept of graph energy, first introduced in 1978, has been a focal point of extensive research within the field of graph theory, leading to the publication of numerous articles. Graph energy, originally associated with the eigenvalues of the adjacency matrix of a graph, has since been extended to various other matrices. These include the maximum degree matrix, Randić matrix, sum-connectivity matrix, and the first and second Zagreb matrices, among others. In this paper, we focus on calculating the energy of several such matrices for the join graph of complete graphs, denoted as J m ( K n ) . Specifically, we compute the energies for the maximum degree matrix, Randić matrix, sum-connectivity matrix, first Zagreb matrix, second Zagreb matrix, reverse first Zagreb matrix, and reverse second Zagreb matrix for J m ( K n ) . Our results provide new insights into the structural properties of the join graph and contribute to the broader understanding of the mathematical characteristics of graph energy for different matrix representations. This work extends the scope of graph energy research by considering these alternative matrix forms, offering a deeper exploration into the algebraic and spectral properties of graph energy in the context of join graphs.

On Sombor index and graph energy of some chemically important graphs

Sombor index of a graph G=(V(G),E(G)) is provided by the expression ∑uv∈E(G)du2+dv2, where dx is the degree of the vertex x∈V(G). The energy of a graph is the quantity given by the total of the absolute values of its adjacency matrix’s eigenvalues. In this article, we improve the relation between the Sombor index and graph energy and derive the relation between them for unicyclic, bicyclic and tricyclic graphs, trees, triangular chain, square cactus chain and hexagonal cactus chain graphs. At last, we find the bounds of graph energy for zigzag and linear hexagonal chains.

On Energy of Prime Ideal Graph of a Commutative Ring Associated with Transmission-Based Matrices

This research explores the energy of the prime ideal graph of a commutative ring. The study demonstrates the energy formula of the graph associated with transmission-based matrices. Through research, the findings highlight the distance, Wiener-Hosoya, and distance signless Laplacian matrices. It should be noted that the distance and Wiener-Hosoya energies are always twice their spectral radius, meanwhile, it does not hold for distance signless Laplacian energy.

Modified reverse degree descriptors for combined topological and entropy characterizations of 2D metal organic frameworks: applications in graph energy prediction.

Topological descriptors are widely utilized as graph theoretical measures for evaluating the physicochemical properties of organic frameworks by examining their molecular structures. Our current research validates the usage of topological descriptors in studying frameworks such as metal-butylated hydroxytoluene, NH-substituted coronene transition metal, transition metal-phthalocyanine, and conductive metal-octa amino phthalocyanine. These metal organic frameworks are crucial in nanoscale research for their porosity, adaptability, and conductivity, making them essential for advanced materials and modern technology. In this study, we provide the topological and entropy characterizations of these frameworks by employing robust reverse degree based descriptors, which offer insightful information on structural complexities. This structural information is applied to predict the graph energy of the considered metal organic frameworks using statistical regression models.

Bounds on Dominating Energy of Graph

Bounds on Dominating Energy of Graph

Bound Inequalities on Degree Sum Energy of Graph

E_ds (G) ,the Degree sum energy of a graph G is the total of all the absolute values of its Degree Sum eigenvalues. In this investigation one upper and lower constraints on the degree Sum energy are obtained in this study.

Signless Laplacian energies of non-commuting graphs of finite groups and related results

The non-commuting graph of a non-abelian group [Formula: see text] with center [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] are adjacent if [Formula: see text]. In this paper, we compute Signless Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of certain finite non-abelian groups. We obtain several conditions such that the non-commuting graph of [Formula: see text] is Q-integral and observe relations between energy, Signless Laplacian energy and Laplacian energy. In addition, we look into the hyperenergetic and hypoenergetic properties of non-commuting graphs of finite groups. We also assess whether the same graphs are Q-hyperenergetic and L-hyperenergetic.

Data-driven approaches to study the spectral properties of chemical structures

The molecular energy, which is the sum of all eigenvalues, is crucial in determining the total π-electron energy of conjugated hydrocarbon molecules. We used machine learning techniques to calculate the energy, inertia, nullity, signature, and Estrada index of molecular graphs for bismuth tri-iodide and benzene rings embedded in P-type surfaces within 2D networks. We applied MATLAB to extract the actual eigenvalues from the data and developed general equations for these molecular properties. We then used these equations to estimate the values and compared them to the actual values through graphical analysis. Our results demonstrate the potential of data-driven techniques in predicting molecular properties and enhancing our understanding of spectral theory.

The Degree Energy of a Graph

The incidence of edges on vertices is a cornerstone of graph theory, with profound implications for various graph properties and applications. Understanding degree distributions and their implications is crucial for analyzing and modeling real-world networks. This study investigates the impact of vertex degree distribution on the energy landscape of graphs in network theory. By analyzing how vertex connectivity influences graph energy, the research enhances the understanding of network structure and dynamics. It establishes important properties and sharp bounds related to degree spectra and degree energy. Furthermore, the study determines the degree spectra and degree energy for several key families of graphs, providing valuable insights with potential applications across various fields.

Predicting graph energy and entropy analysis of pent-heptagonal nanomaterials: Insights from regression models using generalized reverse degree-sum topological indices

Nanostructures have sparked notable interest in materials science due to their diverse applications, especially carbon nanosheets with pentagonal–heptagonal variations have emerged as subjects of intense scientific investigation. These nanostructures show great potential in a wide range of fields, including energy storage, optoelectronics, catalysis, and bioelectronics, thereby making substantial contributions to nanotechnology and material science. Topological indices hold a pivotal position in the domain of mathematical and computational chemistry providing numerical information about molecular structures. This information is instrumental in establishing predictive relationships with chemical and physical properties, finding applications across various fields such as drug development and materials science. This paper introduces a novel generalized version of reverse degree-sum based topological indices that effectively shaping the degree sequence of nanomaterials to fit the physical and chemical properties of dataset, differentiating it from conventional fixed-degree methods. Consequently, we explore entropy-based methodologies to compare the structural complexities of pent-heptagonal nanosheets. Furthermore, our study predicts the graph energies of these nanosheets via linear regression equations, comparing them with energy values predicted by the McClelland equation.

New type bounds for energy of graphs and spread of matrices

The energy of a simple graph G is defined as the sum of the absolute values of eigenvalues of the adjacency matrix of G. For a complex matrix M the spread of M is the maximum absolute value of the differences between any two eigenvalues of M. Thus if λ1,…,λn are the eigenvalues of M, then the spread of M is max1≤i,j≤n⁡|λi−λj|. The spread of a graph G is defined as the spread of its adjacency matrix and is denoted by s(G). The inertia of G is an integer triple (n+,n−,n0) specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix of G. In this paper we find some bounds for energy of graphs in terms of some parameters of graphs such as rank, inertia and spread of graphs. We find some bounds for spread of graphs and matrices that improve the previous bounds. In particular, we show that if G is a graph with m edges and inertia (n+,n−,n0), then s(G)≥2m(n++n−)n+n− and the equality holds if and only if G=rKs∪tK1 or G=rKp,…,p︸q∪tK1 or G=r1Ka1,b1∪⋯∪rhKah,bh, for some non-negative integers r,s,t,p,q and r1,a1,b1,…,rh,ah,bh such that a1b1=⋯=ahbh, p≥2 and q≥3.

Comparative Study on Different Types of Energies

Objectives : The absolute sum of the eigenvalues is the definition of the graph's energy. In addition to discussing their relationship, this study compares the energy of the Adjacency matrix, Laplacian matrix, Signless Laplacian matrix, and Seidel matrix applied to ten distinct kinds of graphs. In this study, a correlation is established between the energy of graphs and the energy of Edge Antimagic graphs. Methods: The technical description of the graph's energy, . An Edge Antimagic graph's energy, identified by (G) = , is the absolute total of its eigenvalues. Findings: The energy was calculated along with its link to the energy of Edge Antimagic matrix, Adjacency matrix, Laplacian matrix, Signless Laplacian matrix, and Seidel matrix. Novelty: Edge antimagic graphs have been applied using the concept of energy. The energy of Edge Antimagic graphs and the energy of graphs were discovered to be related. An analysis was conducted comparing Adjacency energy, Seidel energy, Laplacian energy, and Signless Laplacian energy. Keywords: Laplacian Energy, Spectrum, Seidel Matrix, Signless Laplacian Matrix, Edge Antimagic Graphs.

On the neighbourhood degree sum-based Laplacian energy of graphs

A useful extension of the Laplacian matrix is proposed here and the corresponding modification of the Laplacian energy (LE) is presented. The neighbourhood degree sum-based Laplacian energy (LNE) is produced by means of the eigenvalues of the newly introduced neighbourhood degree sum-based Laplacian matrix (LN). We investigate the mathematical properties of LNE by comparing it with the Laplacian energy. The role of LNE in structure–property modelling of molecules is demonstrated by statistical approach. The formulation of LNE is not ad hoc; rather, its chemical significance exerts that it outperforms LE in modelling physiochemical behaviours of molecules. Mathematical properties of LN are also revealed by finding crucial bounds of its eigenvalues with characterizing extremal graphs.