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In the study of chemical graph theory, an enormous number of research analyses have confirmed that the characteristics of chemicals have a nearby connection with their atomic structure. Topological indices were the critical tools for the analysis of these chemical substances to consider the essential topology of chemical structures. Topological descriptors are the significant numerical quantities or invariant in the fields of chemical graph theory. In this study, we have studied the crystal structure of copper oxide ( Cu 2 O {{\rm{Cu}}}_{2}{\rm{O}} ) chemical graph, and further, we have calculated the ev-degree- and ve-degree-based topological indices of copper oxide chemical graph. This kind of study may be useful for understanding the atomic mechanisms of corrosion and stress–corrosion cracking of copper.

First published online: 13 October 2020 We develop a new robust control scheme for a non-holonomic spherical robot. To this end, the mathematical model of a pendulum driven non-holonomic spherical robot is first presented. Then, a recurrent neural network-based robust nonsingular sliding mode control is proposed for stabilization and tracking control of the system. The designed recurrent neural network is applied to approximate compound disturbances, including external interferences and dynamic uncertainties. Moreover, the controller is designed in a way that avoids the singularity problem in the system. Another advantage of the proposed scheme is its ability for tracking control while there exists control input saturation, which is a serious concern in robotic systems. Based on the Lyapunov theorem, the stability of the closed-loop system has also been confirmed. Lastly, the performance of the proposed control technique for the uncertain system in the presence of an external disturbance, unknown input saturation, and dynamic uncertainties has been investigated. Also, the proposed controller has been compared with a Fuzzy-PID one. Simulation results show the effectiveness and superiority of the developed control technique.

In the present study, a vertically oriented rectangular-shaped enclosure with triangular shaped double fins filled with PCM was considered as a Latent Thermal Energy Storage (LTES) system. Nine different cases including one pure PCM, three nano-PCM (having different volume fractions), two porous-PCM (having different porosity values) and three nano/porous PCM were considered to conduct the investigation. The PCM material employed for investigations was lauric acid. The copper made porous media was assumed to have porosity values of 98% and 95%. The Al 2 O 3 nano-particles with the volume fractions of 1%, 2% and 4% combined with Lauric acid were considered as nano-PCM. The results presented that the cases with nanoparticles volume fractions of 1% and 2% could improve the melting performance. However, the case with nanoparticle volume fraction of 4% deteriorated the melting performance. In the cases with individual usage of nano-PCM the best case was related to the case with the volume fraction of 1% and resulted in about 184s time-saving. It was found that individual usage of porous media could significantly improve the melting performance; however, adding nanoparticles to the cases with porous media, regardless of volume fraction of nanoparticles deteriorates the melting performance of PCM enclosure.

For a graph G , its variable sum exdeg index is defined as SEI a G = ∑ x y ∈ E G a d x + a d y , where a is a real number other than 1 and d x is the degree of a vertex x . In this paper, we characterize all trees on n vertices with first three maximum and first three minimum values of the SEI a index. Also, we determine all the trees of order n with given diameter d and having first three largest values of the SEI a index.

A novel approach to solve optimal control problems dealing simultaneously with fractional differential equations and time delay is proposed in this work. More precisely, a set of global radial basis functions are firstly used to approximate the states and control variables in the problem. Then, a collocation method is applied to convert the time-delay fractional optimal control problem to a nonlinear programming one. By solving the resulting challenge, the unknown coefficients of the original one will be finally obtained. In this way, the proposed strategy introduces a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The algorithm&rsquo, s performance has been analyzed for several non-trivial examples, and the obtained results have shown that this scheme is more accurate, robust, and efficient than most previous methods.

Thermophoretic particle deposition is one of the most rudimentary processes for transferring small particles over a temperature gradient, and it is vital in aero-solution and electronic engineering. In the proposed model, it is intended to explore heat and mass transfer on double-diffusive electrically non-conducting Casson fluid in the two-dimensional flow past a stretching surface with thermophoretic particle deposition. An external magnetic field is subjected to a magnetic dipole. The analysis of Casson fluid along magnetic dipole represents the ferromagnetic fluid. As a consequence of the presence of ferrite particles, the fluid has magnetic properties and is thus a non-Newtonian Casson ferrofluid. Moreover, to enclose heat and mass transfer characteristics, thermal and solutal stratifications are also employed. An eminent bvp4c technique in MATLAB is incorporated to compute these ODEs numerically. Graphical representations of important parameters and their physical behavior are extensively analyzed. The Nusselt number, thermophoretic particle deposition velocity, and drag force coefficient are also addressed graphically. The findings demonstrate that as the values of thermophoretic coefficient and thermophoretic parameter increase, the mass transfer rate diminishes. Further, with the escalating values of the ferromagnetic interaction parameter, the velocity field is reduced but the opposite trend is seen for the thermal field.

The Resistance-Harary index of a connected graph G is defined as R H ( G ) = &sum, { u , v } &sube, V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index.