Energy Of Graph Research Articles

Minimum Injective Eccentric Dominating Energy of Graphs

The minimum injective eccentric dominating energy of a graph is defined. The minimum injective eccentric dominating energy for star, complete and cocktail party graphs are computed. Inspired by McClelland’s bounds for energy of a graph, the upper and lower bounds of minimum injective eccentric dominating energy for star, complete and cocktail party graphs are discussed. Properties of eigenvalues of minimum injective eccentric dominating matrix for star, complete and cocktail party graphs are discussed.

Degree based energy and spectral radius of a graph with self-loops

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of X ⊆ V ( G ) . The general extended adjacency matrix for the graph GX is defined and the bounds for the degree based energy of the graph GX are obtained. The study extends the notion of degree based energy of simple graphs to graphs with self-loops. For the graph GX of order n and size m with σ self-loops, the adjacency energy, E ( G X ) ≥ 4 mn + 2 σ ( n − σ ) n . The spectral radius ρ ( G X ) of its adjacency matrix is always less than or equal to 1 2 [ − 1 + 1 + 4 ( 2 m + σ + Δ − 1 ) ] , where Δ is the maximum degree in the graph GX and the equality conditions are given for 0 < σ < n . Few more bounds for ρ ( G X ) are also obtained. The study shows that, the spectral radius f max ( G X ) of its extended adjacency matrix satisfies: min i ∼ j F ij ( 2 m + σ ) n ≤ f max ( G X ) ≤ max i ∼ j F ij 2 ( 2 m + σ ) . We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.

Exploring spectrum-based descriptors in pharmacological traits through quantitative structure property (QSPR) analysis

The study centered on Quantitative Structure Property Relationship (QSPR) analysis with a focus on various graph energies, investigating drugs like Mefloquinone, Sertraline, Niclosamide, Tizoxanide, PHA-690509, Ribavirin, Emricasan, and Sofosbuvir. Employing computational modeling techniques, the research aimed to uncover the correlations between the chemical structures of these medications and their unique properties. The results illuminated the quantitative relationships between structural characteristics and pharmacological traits, advancing our predictive capabilities. This research significantly contributes to medication discovery and design by providing essential insights into the structure-property connections of these medicinal compounds. Notably, certain spectrum-based descriptors, such as positive inertia energy, adjacency energy, arithmetic-geometric energy, first zegrab energy, and the harmonic index, exhibited strong correlation coefficients above 0.999. In contrast, well-known descriptors like the Extended adjacency, Laplacian and signless Laplacian spectral radii, and the first and second Zagreb Estrada indices showed weaker performance. The article emphasizes the application of graph energies and a linear regression model to predict pharmacological features effectively, enhancing the drug discovery process and aiding in targeted drug design by elucidating the relationship between molecular structure and pharmacological characteristics.

Impact of graph energy on a measurement of resilience for tipping points in complex systems

AbstractSocieties depend on various complex and highly interconnected systems, leading to increasing interest in methods for managing the resilience of these complex systems and the risks associated with their disruption or failure. Identifying and localizing tipping points, or phase transitions, in complex systems is essential for predicting system behavior but a difficult challenge when there are many interacting elements. Systems may transition from stable to unstable at critical tipping‐point thresholds and potentially collapse. One of the suggested approaches in literature is to measure a complex system's resilience to collapse by modeling the system as a network, reducing the network behavior to a simpler model, and then measuring the resulting model's stability. In particular, Gao and colleagues introduced a methodology in 2016 that introduces a resilience index to measure precariousness (the distance to tipping points). However, those mathematical reductions can cause information loss from reducing the topological complexity of the system. Herein, the authors introduce a new methodology that more‐accurately predicts the location of tipping points in networked systems and their precariousness with respect to those tipping points by integrating two approaches: (1) a new measurement of a system's topological complexity using graph energy (created based on molecular orbital theory) and; (2) the resilience index method from Gao et al. This new approach is tested in three separate case studies involving ecosystem collapse, supply chain sustainability, and disruptive technology. Results show a shift in tipping‐point locations correlated with graph energy. The authors present an equation that corrects errors introduced as a result of the model reduction, providing a measurement of precariousness that gives insight into how a complex system's topology affects the location of its tipping points.

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.

Closeness Energy of Non-Commuting Graph for Dihedral Groups

This paper focuses on the non-commuting graph for dihedral groups of order 2n, D2n, where n ≥ 3. We show the spectrum and energy of the graph corresponding to the closeness matrix. The result is that the obtained energy is always twice its spectral radius and is never an odd integer. Moreover, it is classified as hypoenergetic.

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.

Eigenvalues of the generalized subdivision graph with applications to graph energy

For a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], let [Formula: see text] denotes the subdivision graph of [Formula: see text] with vertex set [Formula: see text]. In [Formula: see text], replace each vertex [Formula: see text], [Formula: see text], by [Formula: see text] vertices and join every vertex to the neighbors of [Formula: see text]. Then in the resulting graph, replace each vertex [Formula: see text], [Formula: see text], by [Formula: see text] vertices and join every vertex to the neighbors of [Formula: see text]. The resulting graph is denoted by [Formula: see text]. This generalizes the construction of the subdivision graph [Formula: see text] to [Formula: see text] of a graph [Formula: see text]. In this paper, we provide the complete information about the spectrum of [Formula: see text] using the spectrum of [Formula: see text]. Further, we determine the Laplacian spectrum of [Formula: see text] using the Laplacian spectrum of [Formula: see text], when [Formula: see text] is a regular graph. Also, we find the Laplacian spectrum of [Formula: see text] using the Laplacian spectrum of [Formula: see text] when [Formula: see text]. The energy of a graph [Formula: see text] is defined as the sum of the absolute values of the eigenvalues of [Formula: see text]. The incidence energy of a graph [Formula: see text] is defined as the sum of the square roots of the signless Laplacian eigenvalues of [Formula: see text]. Finally, as an application, we show that the energy of the graph [Formula: see text] is completely determined by the incidence energy of the graph [Formula: see text].

Brick wall diagrams as a completely integrable system

We study the free energy of an integrable, planar, chiral and non-unitary four-dimensional Yukawa theory, the bi-fermion fishnet theory discovered by Pittelli and Preti. The typical Feynman-diagrams of this model are of regular “brick-wall”-type, replacing the regular square lattices of standard fishnet theory. We adapt A. B. Zamolodchikov’s powerful classic computation of the thermodynamic free energy of fishnet graphs to the brick-wall case in a transparent fashion, and find the result in closed form. Finally, we briefly discuss two further candidate integrable models in three and six dimensions related to the brick wall model.

Some Borderenergetic and Equienergetic Graphs

The sum of absolute values of eigenvalues of a graph G is defined as energy of graph. If the energies of two non-isomorphic graphs are same then they are called equienergetic. The energy of complete graph with n vertices is 2(n − 1) and the graphs whose energy is equal to 2(n − 1) are called borderenergetic graphs. It has been revealed that the graphs upto 12 vertices are borderenergetic. It is very challenging and interesting as well to search for borderenergetic graphs with more than 14 vertices. The present work is leap ahead in this direction as we have found a family of borderenergetic graphs of arbitrarily large order. We have also obtained three pairs of equienergetic graphs.

On Second Zagreb Energy of Graphs

Chemical structures are transformed into real numbers employing topological indices, which enable numerical computations to substitute pricey wet lab experiments. The spectral properties of topological indices can be investigated by appropriate modification of adjacency matrix. Gutman and Trinajstic, pioneers in the field of chemical graph theory, presented the Zagreb indices, which have governed topological indices research since 1972. The modified adjacency matrix associated to the second Zagreb index (M2) is studied in this work. The second Zagreb energy (ZE2) is generated from this matrix. The application potential of ZE2 is explored by investigating its efficiency in modelling physico-chemical properties of molecules and isomer discrimination. We aim to set up crucial bounds of spectral radius and spread with identifying extremal graphs. Moreover, the ZE2 energy is investigated for numerous special graphs including regular, semi-regular and chain graphs. We construct a divisor type matrix to estimate the M2-spectrum and ZE2 energy of the chain graph.

Some Remarks on the Randić Energy of Graphs

Some Remarks on the Randić Energy of Graphs

Brain Temporal-Spectral Functional Variability Reveals Neural Improvements of DBS Treatment for Disorders of Consciousness.

Deep brain stimulation (DBS) is establishing itself as a promising treatment for disorders of consciousness (DOC). Measuring consciousness changes is crucial in the optimization of DBS therapy for DOC patients. However, conventional measures use subjective metrics that limit the investigations of treatment-induced neural improvements. The focus of this study is to analyze the regulatory effects of DBS and explain the regulatory mechanism at the brain functional level for DOC patients. Specifically, this paper proposed a dynamic brain temporal-spectral analysis method to quantify DBS-induced brain functional variations in DOC patients. Functional near-infrared spectroscopy (fNIRS) that promised to evaluate consciousness levels was used to monitor brain variations of DOC patients. Specifically, a fNIRS-based experimental procedure with auditory stimuli was developed, and the brain activities during the procedure from thirteen DOC patients before and after the DBS treatment were recorded. Then, dynamic brain functional networks were formulated with a sliding-window correlation analysis of phase lag index. Afterwards, with respect to the temporal variations of global and regional networks, the variability of global efficiency, local efficiency, and clustering coefficient were extracted. Further, dynamic networks were converted into spectral representations by graph Fourier transform, and graph energy and diversity were formulated to assess the spectral global and regional variability. The results showed that DOC patients under DBS treatment exhibited increased global and regional functional variability that was significantly associated with consciousness improvements. Moreover, the functional variability in the right brain regions had a stronger correlation with consciousness enhancements than that in the left brain regions. Therefore, the proposed method well signifies DBS-induced brain functional variations in DOC patients, and the functional variability may serve as promising biomarkers for consciousness evaluations in DOC patients.

Wiener-Hosoya Energy of Non-Commuting Graph for Dihedral Groups

Wiener-Hosoya Energy of Non-Commuting Graph for Dihedral Groups

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.

On some extended energy of graphs and their applications

In this paper, we established a relation between the extended energy of graphs such as the first and second Zagreb energy, Randi?c energy, reciprocal Randi?c energy, and the atom-bond connectivity energy with some thermodynamic properties of benzenoid hydrocarbons. We have seen that these indices are well correlated to the boiling point (BP), Kovats retention index (RI), entropy (S), enthalpy of formation (?Hf), octanol-water partition coefficient (logP), and acentric factor (?) of benzenoid hydrocarbons.

Graph Energy Change Due to Vertex Deletion

Graph Energy Change Due to Vertex Deletion

Some Bounds for The Energies of The Power Graphs of Cyclic Groups

In this study, some lower and upper bounds were obtained for the energies of the power graphs of finite cyclic groups by considering the adjacency matrix structure of the power graph of a finite cyclic group. Then, some results are given using the relationship between the case where the order of a cyclic group is the positive integer power of a prime number and the completeness of the power graph corresponding to this cyclic group.

Demand Prediction of Shared Bicycles Based on Graph Convolutional Network-Gated Recurrent Unit-Attention Mechanism

Shared bicycles provide a green, environmentally friendly, and healthy mode of transportation that effectively addresses the “final mile” problem in urban travel. However, the uneven distribution of bicycles and the imbalance of user demand can significantly impact user experience and bicycle usage efficiency, which makes it necessary to predict bicycle demand. In this paper, we propose a novel shared-bicycle demand prediction method based on station clustering. First, to address the challenge of capturing patterns in station-level bicycle demand, which exhibits significant fluctuations, we employ a clustering method that combines graph information from the bicycle transfer graph and potential energy. This method aggregates closely related stations into corresponding prediction regions. Second, we use the GCN-CRU-AM (Graph Convolutional Network-Gated Recurrent Unit-Attention Mechanism) model to predict bicycle demand in each region. This model extracts the spatial information and correlation between regions, integrates time feature data and local weather data, and assigns weights to the input features. Finally, experimental results based on the data from Citi Bike System in New York City demonstrate that the proposed model achieves a more accurate demand prediction.

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