Search Results (8)

In this work, we consider a generalized form of the classical $(2+1)$-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results. Full article

This paper presents the spectral gap-based topology control algorithm (SGTC) for wireless backhaul networks, a novel approach that employs the Laplacian Spectral Gap (LSG) to find expander-like graphs that optimize the topology of the network in terms of robustness, diameter, energy cost, and network entropy. The latter measures the network’s ability to promote seamless traffic offloading from the Macro Base Stations to smaller cells by providing a high diversity of shortest paths connecting all the stations. Given the practical constraints imposed by cellular technologies, the proposed algorithm uses simulated annealing to search for feasible network topologies with a large LSG. Then, it computes the Pareto front of the set of feasible solutions found during the annealing process when considering robustness, diameter, and entropy as objective functions. The algorithm’s result is the Pareto efficient solution that minimizes energy cost. A set of experimental results shows that by optimizing the LSG, the proposed algorithm simultaneously optimizes the set of desirable topological properties mentioned above. The results also revealed that generating networks with good spectral expansion is possible even under the restrictions imposed by current wireless technologies. This is a desirable feature because these networks have strong connectivity properties even if they do not have a large number of links. Full article

In this paper, we introduce the concepts of Harary Laplacian-energy-like for a simple undirected and connected graph G with order n. We also establish novel matrix results in this regard. Furthermore, by employing matrix order reduction techniques, we derive upper and lower bounds utilizing existing graph invariants and vertex connectivity. Finally, we characterize the graphs that achieve the aforementioned bounds by considering the generalized join operation of graphs. Full article

The structure, energetics, and aromaticity of c.a. 100 constitutional isomers and tautomers of pyrido[m,n]diazepines (m = 1, 2; n = 2, 3, 4, 5; m ≠ n) were studied at the B3LYP/cc-pVTZ level. The pyrido[1,3]diazepines appear the most, while pyrido[2,4]diazepines are the least stable (ca. 26 kcal/mol). In the pyrido[1,n]diazepine group (n = 2–5), the [1,5] isomers are higher in energy by ca. 4.5 kcal/mol and the [1,4] ones by ca. 7 kcal/mol, and the pyrido[1,2]diazepines are the least stable (ca. 20 kcal/mol). All the most stable pyrido[1,n]diazepines have N-atoms near the ring’s junction bond but on opposite sites. The most stable [2,n]-forms are also those with the pyridine ring N6-atom near the junction bond. Surprisingly, for the [1,2]-, [1,3]-, and [1,4]-isomer condensation types of pyridine and diazepine rings, the same N9 > N7 > N6 > N8 stability pattern obeys. The stability remains similar in a water medium simulated with the Polarizable Continuum Model of the solvent and is conserved when calculated using the CAM-B3LYP or BHandHlyp functionals. The ring’s aromaticity in the pyridine[m,n]diazepines was established based on the integral INICS index resulting from the NICSzz-scan curves’ integration. The integral INICS index is physically justified through its relation to the ringcurrent as demonstrated by Berger, R.J.F., et al. Phys. Chem. Chem. Phys. 2022, 24, 624. The six-membered pyrido rings have negative INICS_ZZ indices and can be aromatic only if they are not protonated at the N-atom. All protonated pyrido and seven-membered rings exhibit meaningful positive INICS_ZZ values and can be assigned as antiaromatic. However, some non-protonated pyrido rings also have substantial positive INICS_ZZ indices and are antiaromatic. A weak linear correlation (R² = 0.72) between the INICS_ZZ values of the pyridine I(6) and diazepine I(7) rings exists and is a consequence of the communication between the π-electron systems of the two rings. The juxtaposition of the INICS descriptor of the six- and seven-membered rings and diverse electron density parameters at the Ring Critical Points (RCP) revealed good correlations only with the Electrostatic Potentials from the electrons and nuclei (ESPe and ESPn). The relationships with other RCP parameters like electron density and its Laplacian, total energy, and the Hamiltonian form of kinetic energy density were split into two parts: one nearly constant for the six-membered rings and one linearly correlating for the seven-membered rings. Thus, most of the electron density parameters at the RCP of the six-membered rings of pyridodiazepines practically do not change with the diazepine type and the labile proton position. In contrast, those of the seven-membered rings display aromaticity changes in the antiaromatic diazepine with its ring structural modifications. Full article

Let G be a graph of order n and $L\left(G\right)$ be its Laplacian matrix. The Laplacian polynomial of G is defined as $P(G;\lambda )=det(\lambda I-L\left(G\right))={\sum }_{i=0}^n{(-1)}^i{c}_i\left(G\right){\lambda }^{n-i}$, where $c_i\left(G\right)$ is called the i-th Laplacian coefficient of G. Denoted by ${\mathcal{G}}_{n,m}$ the set of all $(n,m)$-graphs, in which each of them contains n vertices and m edges. The graph G is called uniformly minimal if, for each $i(i=0,1,\dots ,n)$, H is $c_i\left(G\right)$-minimal in ${\mathcal{G}}_{n,m}$. The Laplacian matrix and eigenvalues of graphs have numerous applications in various interdisciplinary fields, such as chemistry and physics. Specifically, these matrices and eigenvalues are widely utilized to calculate the energy of molecular energy and analyze the physical properties of materials. The Laplacian-like energy shares a number of properties with the usual graph energy. In this paper, we investigate the existence of uniformly minimal graphs in ${\mathcal{G}}_{n,m}$ because such graphs have minimal Laplacian-like energy. We determine that the $c_2\left(G\right)$-$c_3\left(G\right)$ successive minimal graph is exactly one of the four classes of threshold graphs. Full article

Cholic acid is a trihydroxy bile acid with a nice peculiarity: the average distance between the oxygen atoms (O7 and O12) of the hydroxy groups located at C7 and C12 carbon atoms is 4.5 Å, a value which perfectly matches with the O/O tetrahedral edge distance in Ih ice. In the solid phase, they are involved in the formation of hydrogen bonds with other cholic acid units and solvents. This fact was satisfactorily used for designing a cholic dimer which encapsulates one single water molecule between two cholic residues, its oxygen atom (Ow) being exactly located at the centroid of a distorted tetrahedron formed by the four steroid hydroxy groups. The water molecule participates in four hydrogen bonds, with the water simultaneously being an acceptor from the 2 O12 (hydrogen lengths are 2.177 Å and 2.114 Å) and a donor towards the 2 O7 (hydrogen bond lengths are 1.866 Å and 1.920 Å). These facts suggest that this system can be a nice model for the theoretical study of the formation of ice-like structures. These are frequently proposed to describe the water structure found in a plethora of systems (water interfaces, metal complexes, solubilized hydrophobic species, proteins, and confined carbon nanotubes). The above tetrahedral structure is proposed as a reference model for those systems, and the results obtained from the application of the atoms in molecules theory are presented here. Furthermore, the structure of the whole system allows a division into two interesting subsystems in which water is the acceptor of one hydrogen bond and the donor of another. The analysis of the calculated electron density is performed through its gradient vector and the Laplacian. The calculation of the complexation energy used correction of the basis set superposition error (BSSE) with the counterpoise method. As expected, four critical points located in the H…O bond paths were identified. All calculated parameters obey the proposed criteria for hydrogen bonds. The total energy for the interaction in the tetrahedral structure is 54.29 kJ/mol, while the summation obtained of the two independent subsystems and the one between the alkyl rings without water is only 2.5 kJ/mol higher. This concordance, together with the calculated values for the electron density, the Laplacian of the electron density, and the lengths of the oxygen atom and the hydrogen atom (involved in the formation of each hydrogen bond) to the hydrogen bond critical point, suggests that each pair of hydrogen bonds can be considered independent of each other. Full article

Cholic acid is a trihydroxy bile acid with three hydroxy groups at C-3, C-7 and C-12 carbon atoms; two methyl groups at C-10 and C-13 carbon atoms of the steroid nucleus; and a carboxylic group at C24 of the side alkyl chain. The distance between the oxygen atoms linked to C-7 and C-12 (~4.5 Å) perfectly matches with the edge distance between oxygen atoms in ice. This leads to the design of a cholic acid dimer in which one water molecule is encapsulated between two cholic residues, resembling an ice-like structure. The water molecule participates in four hydrogen bonds, the water simultaneously being acceptor from the O12-H hydroxy groups (two bonds with lengths of 2.177 Å and 2.114 Å) and the donor towards the O-7-H groups (two bonds with lengths of 1.866 Å and 1.920 Å). Regarding this communication, we present the application of the “atoms in molecules” (AIM) theory to the tetrahedral structure. The analysis of the calculated electron density, ρ, is performed using its gradient vector, ∇ρ, and the Laplacian, ∇²ρ. The calculation of the complexation energy used correction of the basis set superposition error (BSSE) and the counterpoise method. As expected, four critical (3,−1) points located in the H^…O bond paths were identified. All calculated parameters are in concordance with those of similar systems and obey the proposed criteria for hydrogen bonds. The total energy for the interaction is −12.67 kcal/mol and is analysed using proposed energy/electron density equations. Full article

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than $m$ . Full article