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Carbon nanotubes are one of the most extensively studied nanomaterials because of their remarkable mechanical and electrical properties. The Y-junction structures within carbon nanotubes have received significant attention in the field of nanotechnology, primarily due to their immense potential for powering the next generation of multi-terminal nanodevices. Topological indices play a crucial role in exploring the physicochemical properties and structural attributes of chemical compounds, as they are numerical values intricately linked to the molecular structure of these compounds. Moreover, graph-based entropies serve as essential thermophysical parameters used to quantify the heterogeneity and relative stabilities of molecular structures. In this article, we have utilized the NM-polynomial technique to calculate various neighborhood degree sum-based topological indices and graph-based entropies for carbon nanotube Y-junction graphs.

We investigate novel transport properties of chiral continuous-time quantum walks (CTQWs) on graphs. By employing a gauge transformation, we demonstrate that CTQWs on chiral chains are equivalent to those on non-chiral chains, but with additional momenta from initial wave packets. This explains the novel transport phenomenon numerically studied in (Khalique et al 2021 New J. Phys. 23 083005). Building on this, we delve deeper into the analysis of chiral CTQWs on the Y-junction graph, introducing phases to account for the chirality. The phase plays a key role in controlling both asymmetric transport and directed complete transport among the chains in the Y-junction graph. We systematically analyze these features through a comprehensive examination of the chiral CTQW on a Y-junction graph. Our analysis shows that the CTQW on Y-junction graph can be modeled as a combination of three wave functions, each of which evolves independently on three effective open chains. By constructing a lattice scattering theory, we calculate the phase shift of a wave packet after it interacts with the potential-shifted boundary. Our results demonstrate that the interplay of these phase shifts leads to the observed enhancement and suppression of quantum transport. The explicit condition for directed complete transport or 100 % efficiency is analytically derived. Our theory has applications in building quantum versions of binary tree search algorithms.

Carbon nanotube Y-shaped junctions (normally called as Y-junctions) are constructed by inserting heptagons into the graphene sheet. The design requires the inclusion of at least 6 heptagons at the junction where 3 carbon nanotubes joined. With the growing focus on carbon nanotubes, their junctions have garnered increased attention for their applications in various scientific fields. Chemical structures can be expressed in graphs, where atoms represent vertices, and the bonds between the atoms are called edges. To obtain the exact position of an atom, which is unique from all the atoms, several atoms are selected, this is called resolving set. The minimum number of atoms in the resolving set is called the metric dimension. In this paper, we have computed the metric dimension of carbon nanotube Y-junctions, assigning each atom a unique identifier to facilitate precise location. The metric dimension is constant for all the values of the 3 parameters included to develop a Y-junction. It resulted in 3 metric dimensions for the entire Y-junction. It means that whatever the order and quantity of nanotubes attached to it, the metric dimension will remain constant with number 3.

Recent research on nanostructures has demonstrated their importance and application in a variety of fields. Nanostructures are used directly or indirectly in drug delivery systems, medicine and pharmaceuticals, biological sensors, photodetectors, transistors, optical and electronic devices, and so on. The discovery of carbon nanotubes with Y-shaped junctions is motivated by the development of future advanced electronic devices. Because of their interaction with Y-junctions, electronic switches, amplifiers, and three-terminal transistors are of particular interest. Entropy is a concept that determines the uncertainty of a system or network. Entropy concepts are also used in biology, chemistry, and applied mathematics. Based on the requirements, entropy in the form of a graph can be classified into several types. In 1955, graph-based entropy was introduced. One of the types of entropy is edge-weighted entropy. We examined the abstract form of Y-shaped junctions in this study. Some edge-weight-based entropy formulas for the generic view of Y-shaped junctions were created, and some edge-weighted and topological index-based concepts for Y-shaped junctions were discussed in the present paper.

The current results of various forms of carbon nanostructures and its applications in different areas attract the researchers. In pharmaceutical, medicine, industry and electronic devices they used it by its graphical invariants. The detection of different types of carbon nanotubes junctions enhanced the attention and interest for forthcoming devices like transistors and amplifiers. A topological index plays a very important role in the study of physicochemical properties of biological and chemical structures. In this paper, we determine results of ve-degree topological indices for various type of carbon nanotubes Y-junctions and their comparisons. The particular indices called as The first ve-degree Zagreb β index, the second ve-degree Zagreb index, ve-degree Randic index, ve-degree atom-bond connectivity index, ve-degree geometric-arithmetic index, ve-degree harmonic index and ve-degree sum-connectivity index.

The aim of this work is to establish a linear instability result of static, kink and kink/anti-kink soliton profile solutions for the sine-Gordon equation on a metric graph with a structure represented by a Y -junction. The model considers boundary conditions at the graph-vertex of δ -interaction type. It is shown that kink and kink/anti-kink soliton type static profiles are linearly (and nonlinearly) unstable. For that purpose, a linear instability criterion that provides the sufficient conditions on the linearized operator around the wave to have a pair of real positive/negative eigenvalues, is established. As a result, the linear stability analysis depends upon the spectral study of this linear operator and of its Morse index. The extension theory of symmetric operators, Sturm–Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. A comprehensive study of the local well-posedness of the sine-Gordon model in E ( Y ) × L 2 ( Y ) where E ( Y ) ⊂ H 1 ( Y ) is an appropriate energy space, is also established. The theory developed in this investigation has prospects for the study of the instability of static wave solutions of other nonlinear evolution equations on metric graphs.

We report on high-precision measurements that were performed with superconducting waveguide networks with the geometry of a tetrahedral and a honeycomb graph. They consist of junctions of valency three that connect straight rectangular waveguides of incommensurable lengths. The experiments were performed in the frequency range of a single transversal mode, where the associated Helmholtz equation is effectively one dimensional and waveguide networks may serve as models of quantum graphs with the joints and waveguides corresponding to the vertices and bonds. The tetrahedral network comprises T junctions, while the honeycomb network exclusively consists of Y junctions, that join waveguides with relative angles 90 degree and 120 degree, respectively. We demonstrate that the vertex scattering matrix, which describes the propagation of the modes through the junctions strongly depends on frequency and is non-symmetric at a T junction and thus differs from that of a quantum graph with Neumann boundary conditions at the vertices. On the contrary, at a Y junction, similarity can be achieved in a certain frequeny range. We investigate the spectral properties of closed waveguide networks and fluctuation properties of the scattering matrix of open ones and find good agreement with random matrix theory predictions for the honeycomb waveguide graph.

We investigate novel transport properties of chiral continuous-time quantum walks (CTQWs) on graphs. By employing a gauge transformation, we demonstrate that CTQWs on chiral chains are equivalent to those on non-chiral chains, but with additional momenta from initial wave packets. This explains the novel transport phenomenon numerically studied in [New J. Phys. 23, 083005(2021)]. Building on this, we delve deeper into the analysis of chiral CTQWs on the Y-junction graph, introducing phases to account for the chirality. The phase plays a key role in controlling both asymmetric transport and directed complete transport among the chains in the Y-junction graph. We systematically analyze these features through a comprehensive examination of the chiral continuous-time quantum walk (CTQW) on a Y-junction graph. Our analysis shows that the CTQW on Y-junction graph can be modeled as a combination of three wave functions, each of which evolves independently on three effective open chains. By constructing a lattice scattering theory, we calculate the phase shift of a wave packet after it interacts with the potential-shifted boundary. Our results demonstrate that the interplay of these phase shifts leads to the observed enhancement and suppression of quantum transport. The explicit condition for directed complete transport or 100% efficiency is analytically derived. Our theory has applications in building quantum versions of binary tree search algorithms.

We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Soliton solutions of the nonlinear Dirac equation on simple metric graphs are obtained. It is shown that these solutions provide reflectionless vertex transmission of the Dirac solitons under suitable conditions. The constraints for bond nonlinearity coefficients, allowing reflectionless transmission over a Y-junction are derived. The analytical results are confirmed by direct numerical simulations.

In the present paper an introduction to the new subject of nonlinear dispersive hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of nonlinear Schr\"odinger equation. Special consideration is given to existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning NLS equation on a star graph: the standing waves of NLS equation on a graph with a \\(\\delta\\) interaction at the vertex; the scattering of fast solitons through an Y-junction in the cubic case. The emphasis is on description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.