Narrow Results

The non-commuting graph of a non-abelian group $G$ with center $Z(G)$ is a simple undirected graph whose vertex set is $G\setminus Z(G)$ and two vertices $x,y$ are adjacent if $xy\ne yx$. 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 $G$ 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.

In this study, we introduce a novel index called the modified inverse sum indeg (MISI) energy as an extension of the traditional inverse sum indeg (ISI) energy and neighbor degree sum energy. We devised an algorithm that employs the simplified molecular input line entry system (SMILES) formula of compounds to compute the MISI energy of polycyclic aromatic hydrocarbon (PAH). Additionally, we established the bounds of the MISI energy for certain classes of graphs with fixed number of vertices. A strong correlation between the MISI energy and the total $\pi$-electron energy of PAHs is observed through regression analysis. Remarkably, this correlation outperforms that achieved by the classical ISI energy. These findings highlight the enhanced effectiveness of the MISI energy as a descriptor for capturing significant molecular properties. Moreover, our study establishes a foundation for further theoretical extensions and practical applications in the field of chemical graph theory.

For a graph $G$ with $n$ vertices and $m$ edges, the energy (or the adjacency energy) is the sum of the absolute values of the eigenvalues (or adjacency eigenvalues) of $G$, that is, ${ℰ}(G)={\sum }_{i=1}^n|{\lambda }_i(G)|$, where ${\lambda }_i(G)$ is the $i\mathrm{\text{th}}$ adjacency eigenvalue of $G$. In this paper, we discuss the characteristic polynomial and the energy of dendrimers. We first show the formula obtained for the characteristic polynomial and the energy of tree dendrimers considered in Bokhary and Tabassum [The energy of some tree dendrimers, J. Appl. Math. Comput.68 (2022) 1033–1045] are incorrect. Using a different algorithmic procedure we will derive the correct formula for the characteristic polynomial and the energy of dendrimer $d(3,k)$. Further, we obtain the energy of $d(l,k)$ for $k=4$ and $k=5$, which partially answers a problem raised in Bokhary and Tabassum [The energy of some tree dendrimers, J. Appl. Math. Comput.68 (2022) 1033–1045].

We define a scale-invariant energy function for polygonal knots in ℜ³ based on the minimum distances between segments. The energy is bounded below by 2π. (minimum crossing number of the knot type). For each knot type, there exists an ideal number of segments, from which can be made an ideal conformation of the knot having minimum energy among all polygons realizing that knot type. Results leading to this include the following: The energy of an n-segment polygon is greater than n; if energy is bounded then ratios of edge lengths and angles are bounded away from zero; changing knot type requires passing an infinite energy barrier. We implement an algorithm, with the feature that preserving knot-type is guaranteed, to discover local energy-minimizing conformations for a given number of segments, and present some of these examples.

The stochastic wave equation in one spatial dimension driven by a class of fractional noises or, alternately, by a class of smooth noises with arbitrary temporal covariance is studied. In either case, the wave equation is explicitly solved and the upper and lower bounds on both the large and small deviations of several sup norms associated with the solution are given. Finally the energy of a system governed by such an equation is calculated and its expected value is found.

Let ${\mathcal{𝒢}}_R$ denote the unitary homogeneous bi-Cayley graph over a finite commutative ring $R$. In this paper, we determine the energy of ${\mathcal{𝒢}}_R$ and that of its complement and line graph, and characterize when such graphs are hyperenergetic. We also give a necessary and sufficient condition for ${\mathcal{𝒢}}_R$ (respectively, the complement of ${\mathcal{𝒢}}_R$, the line graph of ${\mathcal{𝒢}}_R$) to be Ramanujan.

In this paper, image fusion method based on a new class of wavelet — non-separable wavelet with compactly supported, linear phase, orthogonal and dilation matrix is presented. We first construct a non-separable wavelet filter bank. Using these filters, the images involved are decomposed into wavelet pyramids. Then the following fusion algorithm was proposed: for low-frequency part, the average value is selected for new pixel value, For the three high-frequency parts of each level, the standard deviation of each image patch over 3×3 window in the high-frequency sub-images is computed as activity measurement. If the standard deviation of the area 3×3 window is bigger than the standard deviation of the corresponding 3×3 window in the other high-frequency sub-image. The center pixel values of the area window that the weighted area energy is bigger are selected. Otherwise the weighted value of the pixel is computed. Then a new fused image is reconstructed. The performance of the method is evaluated using the entropy, cross-entropy, fusion symmetry, root mean square error and peak-to-peak signal-to-noise ratio. The experiment results show that the non-separable wavelet fusion method proposed in this paper is very close to the performance of the Haar separable wavelet fusion method.

The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction, we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e. the details of how within fourth order gravity with L= R + R², the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar-tensor theory, will be shortly presented.

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular, we prove that their relative "energy" (defined as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quantities related to a Lie subalgebra of vector fields isomorphic to the Poincaré algebra. These quantities are also conserved in time. We also find a relative conserved quantity for such a kind of solution which is conserved in time though non-vanishing. This example provides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter fluid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our superpotential (that happens to coincide with the so-called KBL potential although it is generated differently) provides strong conservation laws under much weaker hypotheses than the ones usually required. In particular, the symmetry generator is not needed to be Killing (nor Killing of the background, nor asymptotically Killing), the prescription is quasi-local and it works fine in a finite region too and no matching condition on the boundary is required.

In this study, we investigate the special type of magnetic trajectories associated with a magnetic field ${ℬ}$ defined on a 3D Riemannian manifold. First, we consider a moving charged particle under the action of a frictional force, $f$, in the magnetic field ${ℬ}$. Then, we assume that trajectories of the particle associated with the magnetic field ${ℬ}$ correspond to frictional magnetic curves ($f$-magnetic curves$)$ of magnetic vector field ${ℬ}$ on the 3D Riemannian manifold. Thus, we are able to investigate some geometrical properties and physical consequences of the particle under the action of frictional force in the magnetic field ${ℬ}$ on the 3D Riemannian manifold.

In this paper, we study a special type of magnetic trajectories associated with a magnetic field ${ℬ}$ defined on a 3D Riemannian manifold. First, we assume that we have a moving charged particle which is supposed to be under the action of a gravitational force $G$ in the magnetic field ${ℬ}$ on the 3D Riemannian manifold. Then, we determine trajectories of the charged particle associated with the magnetic field ${ℬ}$ and we define gravitational magnetic curves ($G$-magnetic curves) of the magnetic vector field ${ℬ}$ on the 3D Riemannian manifold. Finally, we investigate some geometrical and physical features of the moving charged particle corresponding to the $G$-magnetic curve. Namely, we compute the energy, magnetic force, and uniformity of the $G$-magnetic curve.

In this paper, we show that the velocity vector field of classical Bernoulli–Euler elastic curve is harmonic in $R^n$ space. We propose a new characterization for classical Bernoulli–Euler elastic curves and plot graphs of examples that satisfy this characterization.

In this study, we firstly characterize modified magnetic particles and associated magnetic fields by considering the evolution of the charged particle together with its modified frame construction in 3D space. Then, we compute the energy flows of each modified magnetic particles in terms of its curvature and torsion in the space. Finally, we obtain energy flux density, the intensity of the light, power of modified magnetic particles in the space.

The more precise definition and the more fundamental understanding of the concepts of time, energy, entropy and information are building upon the new, relativistic foundation of gravity. This lecture is an attempt to explain the basic principles that underpin this progress, by focusing on the simple but subtle universal definition of energy. The principles are unearthed from Einstein’s theory and Noether’s theorems, beneath a century of misconceptions.

In this work, we calculate new concept for their Fermi–Walker conformable derivatives with spherical timelike magnetic fiber in spherical de-Sitter space ${𝕊}_1^2$. The description is advanced for timelike magnetic trajectory. First, we find conformable fractional derivatives of spherical magnetic Lorentz fields. Moreover, we compute the Fermi–Walker conformable derivatives of the normalization and recursion operators for these spherical vector fields. Furthermore, we give some characterizations of Lorentz fields of electromagnetic fields of associated with spherical magnetic conformable particle. Finally, we obtain the energy of the conformable derivative of the normalization function of the Lorentz forces and we have energy relations of some vector fields associated with spherical frame in the de-Sitter space ${𝕊}_1^2$.

The energy of a weighted digraph $D$ is the sum of absolute values of real part of its eigenvalues. Recently, the minimal and maximal energy of unicyclic weighted digraphs with cycle weight $r\in [-1,1]\setminus \{0\}$ is studied. In this paper, we introduce a class $𝔹(r,s)$ of those bicyclic weighted digraphs of fixed order which contain vertex-disjoint weighted cycles of weights $r$ or $-r$ and $s$ or $-s$, where $0<r,s\le 1$. We find digraphs in $𝔹(r,s)$ under certain conditions on $r$ and $s$ with extremal energy.

The energy of a square matrix is the sum of the absolute values of deviations of its eigenvalues from their mean. In this paper, we extend the notion of the Laplacian and the signless Laplacian energy of a graph to non-negative real symmetric matrices with zero diagonal. Such a matrix $\widehat{A}$ can be considered as the adjacency matrix of some weighted graph. Considering the sum of the weights of edges that are incident to a vertex as the weight of the corresponding vertex, we construct the diagonal matrix $\widehat{D}$ of which the diagonal entries are the weights of the corresponding vertices. We define a weighted version of the Laplacian matrix as $\widehat{L}=\widehat{D}-\widehat{A}$ and that of the signless Laplacian matrix as $\widehat{Q}=\widehat{D}+\widehat{A}$. In this work, we obtain some bounds of the energies of the weighted adjacency matrix, the weighted Laplacian matrix, and the weighted signless Laplacian matrix by studying the corresponding weighted graph. We also obtain some relations interconnecting those energies.

A mixed graph is a graph whose edge set consists of both oriented and unoriented edges. The Hermitian-adjacency matrix of an $n$-vertex mixed graph is a square matrix $H(M)=[h_{jk}]$ of order $n$, where $h_{jk}=\dot{\iota }=-h_{kj}$ if there is an arc from $v_j$ to $v_k$ and $h_{jk}=1$ if there is an edge between $v_j$ and $v_k$, and $h_{jk}=0$ otherwise. Let $D(M)=[d_{jj}]$ be a diagonal matrix, where $d_{jj}$ is the degree of $v_j$ in the underlying graph of $M$. The matrices $L(M)=D(M)-H(M)$ and $Q(M)=D(M)+H(M)$ are, respectively, the Hermitian Laplacian and Hermitian quasi-Laplacian matrix of the mixed graph $M$. In this paper, we first found coefficients of the characteristic polynomial of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph $M$. Second, we discussed relationship between the spectra of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph $M$.

Let ${ℛ}$ be commutative ring and $Z^{\ast }({ℛ})$ be the set of all non-zero zero divisors of ${ℛ}$. Then $\mathrm{\Gamma }({ℛ})$ is said to be the zero divisor graph if and only if $a\cdot b=0$ where $a,b\in V(\mathrm{\Gamma }({ℛ}))$ and $(a,b)\in E(\mathrm{\Gamma }({ℛ}))$. Graph energy ${ℰ}(\mathrm{\Gamma }({ℛ}))$ is defined as the sum of the absolute eigenvalues of the adjacency matrix ${ℳ}(\mathrm{\Gamma }({ℛ}))$, then ${ℰ}(\mathrm{\Gamma }({ℛ}))={\sum }_{i=1}^n\left|{\lambda }_i\right|$. Wiener index $W(\mathrm{\Gamma }({ℛ}))$ is defined as the sum of all distance between pairs of vertices $a$ and $b$, then $W(\mathrm{\Gamma }({ℛ}))={\sum }_{a,b\in V(\mathrm{\Gamma }({ℛ}))}d(a,b)$. 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.

The distance signless Laplacian spectral radius of a connected graph $G$ is the largest eigenvalue of the distance signless Laplacian matrix of $G$, defined as $D^Q(G)=\mathrm{\text{Tr}}(G)+D(G)$, where $D(G)$ is the distance matrix of $G$ and $\mathrm{\text{Tr}}(G)$ is the diagonal matrix of vertex transmissions of $G$. In this paper, we determine some bounds on the distance signless Laplacian spectral radius of $G$ based on some graph invariants, and characterize the extremal graphs. In addition, we define distance signless Laplacian energy, similar to that in [J. Yang, L. You and I. Gutman, Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math.37 (2013) 245–255] and give some bounds on the distance signless Laplacian energy of graphs.