Energy Of Graph Research Articles

Characteristic polynomials, spectral-based Riemann-Zeta functions and entropy indices of n-dimensional hypercubes.

We obtain the characteristic polynomials and a number of spectral-based indices such as the Riemann-Zeta functional indices and spectral entropies of n-dimensional hypercubes using recursive Hadamard transforms. The computed numerical results are constructed for up to 23-dimensional hypercubes. While the graph energies exhibit a J-curve as a function of the dimension of the n-cubes, the spectra-based entropies exhibit a linear dependence on the dimension. We have also provided structural interpretations for the coefficients of the characteristic polynomials of n-cubes and obtain expressions for the integer sequences formed by the spectral-based Riemann-Zeta functions.

Some upper and lower bounds for $D_{\alpha}$-energy of graphs

The generalized distance matrix of a connected graph $G$, denoted by $D_{\alpha}(G)$, is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G), ~~~~ 0\leq \alpha\leq 1$. Here, $D(G)$ is the distance matrix and $Tr(G)$ represents the vertex transmissions. Let $\partial_{1}\geq \partial_{2}\geq \cdots \geq \partial_{n}$ be the eigenvalues of $D_{\alpha}(G)$ and let $W(G)$ be the Wiener index. The generalizeddistance energy of $G$ can be defined as $E^{D_{\alpha}}(G)=\displaystyle\sum_{i=1}^{n}\left|\partial_i-\frac{2\alpha W(G)}{n}\right|$. In this paper, we develop some new theory regarding the generalized distance energy $E^{D_{\alpha}}(G)$ for a connected graph $G$. We obtain some sharp upper and lower bounds for$E^{D_{\alpha}}(G)$ connecting a wide range of parameters in graph theory including the maximum degree $\Delta$, the Wiener index $W(G)$, the diameter $d$, the transmission degrees, and the generalized distance spectral spread $D_{\alpha}S(G)$. We characterized the special graph classes that attain the bounds.

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.

Some Results on Spectrum and Energy of Graphs with Loops

Some Results on Spectrum and Energy of Graphs with Loops

Some new results on energy of graphs with self loops

The graph $$G_\sigma $$ is obtained from graph G by attaching self loops on $$\sigma $$ vertices. The energy $$ E(G_\sigma )$$ of the graph $$G_\sigma $$ with order n and eigenvalues $$\lambda _1,\lambda _2,\dots ,\lambda _n$$ is defined as $$ E(G_\sigma )= \displaystyle \sum _{i=1}^n\bigg |\lambda _i-\dfrac{\sigma }{n}\bigg |$$ . It has been proved that if $$\sigma =0\; or\; n$$ then $$ E(G)=E(G_\sigma ) $$ . The obvious question arise: Are there any graph such that $$E(G)=E(G_\sigma )$$ and 0 $$<\sigma <n$$ ? We have found an affirmative answer of this question and contributed a graph family which satisfies this property.

Graph energy and topological descriptors of zero divisor graph associated with commutative ring

Graph energy and topological descriptors of zero divisor graph associated with commutative ring

Sombor index and eigenvalues of comaximal graphs of commutative rings

The comaximal graph [Formula: see text] of a commutative ring [Formula: see text] is a simple graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we find the sharp bounds for the Sombor index for comaximal graphs of integer modulo ring [Formula: see text] and give the corresponding extremal graphs. Also, we find the Sombor eigenvalues and the bounds for the Sombor energy of comaximal graphs of [Formula: see text].

Weighted graphs with an application to returned sequence

Its well known that there are a limited number of method which associate group theory to graph theory. In this paper, we propose a new method and obtain a novel kind of graphs. Some important application like graph energy was established to the proposed graph.

Quotient Energy of Zero Divisor Graphs And Identity Graphs

Consider the (p,q) simple connected graph . The sum absolute values of the spectrum of quotient matrix of a graph make up the graph's quotient energy. The objective of this study is to examine the quotient energy of identity graphs and zero-divisor graphs of commutative rings using group theory, graph theory, and applications. In this study, the identity graphs derived from the group and a few classes of zero-divisor graphs of the commutative ring R are examined.

Knowledge discovery assistants for crash simulations with graph algorithms and energy absorption features

We propose the representation of data from finite element car crash simulations in a graph database to empower analysis approaches. The industrial perspective of this work is to narrow the gap between the uptake of modern machine learning methods and the current computer-aided engineering-based vehicle development workflow. The main goals for the graph representation are to achieve searchability and to enable pattern and trend investigations in the product development history. In this context, we introduce features for car crash simulations to enrich the graph and to provide a summary overview of the development stages. These features are based on the energy output of the finite element solver and, for example, enable filtering of the input data by identifying essential components of the vehicle. Additionally, based on these features, we propose fingerprints for simulation studies that assist in summarizing the exploration of the design space and facilitate cross-platform as well as load-case comparisons. Furthermore, we combine the graph representation with energy features and use a weighted heterogeneous graph visualization to identify outliers and cluster simulations according to their similarities. We present results on data from the real-life development stages of an automotive company.

The Extended Adjacency Indices for Several Types of Graph Operations

Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a simple graph G, the extended adjacency index <I>EA(G)</I> is also defined by some researchers. All of them play important roles in mathematics and chemistry. In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product. In addition, we also give some examples of different combinations of special graphs, such as complete graphs and cycle graphs, and the classical graph, Cayley graph. By combining the special structure of the graph, it will pave the way for the calculation of some chemical or biological classical molecular structure. We can find that it plays a meaningful role in calculating the structure of complex chemical molecules through the graph operation of EA index on the any simple combined graphs, and it can also play a role in biology, physics, medicine and so on. Finally, we put forward some other related problems that can be further studied in the future.

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 Randić energy of graphs

Let di be the degree of vertex vi of G then Randić matrix R(G) = [rij ] is defined as rij = 1/ √didj, if the vertices vi and vj are adjacent in G or rij = 0, otherwise. The Randić energy is the sum of absolute values of the eigenvalues of R(G). In this paper we have investigated Randić energy of m-Splitting and m-Shadow graphs. We also have constructed a sequence of graphs having same Randić energy.

Smoothness of Graph Energy in Chemical Graphs

The energy of a graph G as a chemical concept leading to HMO theory was introduced by Hückel in 1931 and developed into a mathematical interpretation many years later when Gutman in 1978 gave his famous definition of the graph energy as the sum of the absolute values of the eigenvalues of the adjacency matrix of G. One of the general requirements for any topological index is that similar molecules have close TI values, which is called smoothness. To explore this property, we consider two variants of structure sensitivity and abruptness as introduced by Furtula et al. in 2013 and 2019, for hydrocarbons with up to 20 carbon atoms. Finally, we investigate the relationships between graph energies of acyclic hydrocarbons compared to their cyclic versions.

Distance Laplacian spectral ordering of sun type graphs

Let G be a simple, connected graph of order n. Its distance Laplacian energy DLE(G) is given by DLE(G)=∑i=1n|ρiL−2W(G)n|, where ρ1L≥ρ2L≥⋯≥ρnL are the distance Laplacian eigenvalues and W(G) is the Wiener index of G. Distance Laplacian eigenvalues of sun and partial sun graphs have been characterized. We order the partial sun graphs by using their second largest distance Laplacian eigenvalue. Moreover, the distance Laplacian energy of sun and partial sun graphs have been derived in this paper. These graphs are also ordered by using their distance Laplacian energies.

Detection of healthy and pathological heartbeat dynamics in ECG signals using multivariate recurrence networks with multiple scale factors

The electrocardiogram (ECG) is one of the physiological signals applied in medical clinics to determine health status. The physiological complexity of the cardiac system is related to age, disease, etc. For the investigation of the effects of age and cardiovascular disease on the cardiac system, we then construct multivariate recurrence networks with multiple scale factors from multivariate time series. We propose a new concept of cross-clustering coefficient entropy to construct a weighted network, and calculate the average weighted path length and the graph energy of the weighted network to quantitatively probe the topological properties. The obtained results suggest that these two network measures show distinct changes between different subjects. This is because, with aging or cardiovascular disease, a reduction in the conductivity or structural changes in the myocardium of the heart contributes to a reduction in the complexity of the cardiac system. Consequently, the complexity of the cardiac system is reduced. After that, the support vector machine (SVM) classifier is adopted to evaluate the performance of the proposed approach. Accuracy of 94.1% and 95.58% between healthy and myocardial infarction is achieved on two datasets. Therefore, this method can be adopted for the development of a noninvasive and low-cost clinical prognostic system to identify heart-related diseases and detect hidden state changes in the cardiac system.

Some Relations Between Rank, Vertex Cover Number and Energy of Graph

In this paper, we extend some results of [F. Shaveisi, lower bounds on the vertex cover number and energy of graphs, MATCH Commun. Math. Comput. Chem, 87(3) (2022) 683-692] which state some relations between the vertex cover and other parameters, such as the order and maximum or minimum degree of graphs. Also, we prove that for a graph G, E(G) ≥ 2β(G)−2Ce(G) and so E(G) ≥ 2β(G) − 2C(G), where E(G), β(G), Ce(G) and C(G) denote the energy, vertex cover, number of even cycles and number of cycles in G, respectively. For these both inequalities we investigate their equality. Finally, we give some relations between E(G),γ(G) and γt(G), where γ(G) and γt(G) are domination number and total domination number of G, respectively.

Spectrum of prism graph and relation with network related quantities

&lt;abstract&gt;&lt;p&gt;Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius.&lt;/p&gt;&lt;/abstract&gt;

Correlation Coefficient Measure of Intuitionistic Fuzzy Graphs with Application in Money Investing Schemes

Intuitionistic fuzzy graphs are extensions of fuzzy graphs that preserve the dualism characteristics of fuzzy graphs and have a stronger capacity to describe ambiguity in actual decision-making issues than fuzzy graphs. In this research paper, the Laplacian energy and correlation coefficient of intuitionistic fuzzy graphs are computed for finding group decision-making problems that are supported by intuitionistic fuzzy preference relations. We propose a novel method for calculating establishments' comparative position loads by manipulating the undecided corroboration of IFPR and the correlation coefficient of one personality IFPR to the other items. As a result, we comprehend a large number of establishments in the detailed IFPR and devise a correlation coefficient process to investigate the significance of alternatives and the best of the alternatives. Finally, we present a collaborative decision-making technique in a money-investing scheme, and that idea may be devised in disparate beneficial investing schemes.

Minimum mean covering Gutman energy of certain graphs

The ᶆean-energy orᵯ-energy of a graph was introduced by Laura Buggy, Amaliailiuc, Katelyn Mc.Call, Duyguyen in 2010. The minimum mean covering energy (MMCE) is a special type of graph energy, symbolically represented by ( ), c E G introduced by C. Adiga et. al. The sum of the magnitudes of the characteristic values obtained from the Gutman matrix is named as Gutman energy. we present the results of minimum mean covering Gutman energies (MMCGE) of certain standard graphs (())MCGEG and the lower and upper bounds of ()MCGEG were also established.