Analysts are creating materials, for example, a carbon nanotube-based composite created by NASA that bends when a voltage is connected. Applications incorporate the use of an electrical voltage to change the shape (transform) of airship wings and different structures. Topological indices are numerical parameters of a molecular graph which characterize its topology and are usually graph invariant. Topological indices are used, for example, in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. Topological indices catch symmetry of molecular structures and help us to predict properties, for example, boiling points, viscosity, and the radius of gyrations of nanotubes. In this paper, we compute M-polynomials of two nanotubes, VC ₅ C ₇ [p, q] and HC ₅ C ₇ [p, q]. By applying calculus on these M-polynomials, we produce formulas of numerous degree-based topological indices, which are functions relying on parameters of the structure and, in combination, decide properties of the concerned polymeric compound.

The symmetry design of the system contains integer partial differential equations and fractional-order partial differential equations with fractional derivative. In this paper, we develop a scheme to examine fractional-order shock wave equations and wave equations occurring in the motion of gases in the Caputo sense. This scheme is formulated using the Mohand transform (MT) and the homotopy perturbation method (HPM), altogether called Mohand homotopy perturbation transform (MHPT). Our main finding in this paper is the handling of the recurrence relation that produces the series solutions after only a few iterations. This approach presents the approximate and precise solutions in the form of convergent results with certain countable elements, without any discretization or slight perturbation theory. The numerical findings and solution graphs attained using the MHPT confirm that this approach is significant and reliable.

The algebras of the symmetry operators for the Klein–Gordon equation are important for a charged test particle, moving in an external electromagnetic field in a space time manifold on the isotropic hydrosulphate. In this paper, we develop an analytical and numerical approach for providing the solution to a class of linear and nonlinear fractional Klein–Gordon equations arising in classical relativistic and quantum mechanics. We study the Yang homotopy perturbation transform method ($\mathbb{Y}$HPTM), which is associated with the Yang transform ($\mathbb{Y}$T) and the homotopy perturbation method (HPM), where the fractional derivative is taken in a Caputo–Fabrizio (CF) sense. This technique provides the solution very accurately and efficiently in the form of a series with easily computable coefficients. The behavior of the approximate series solution for different fractional-order ℘ values has been shown graphically. Our numerical investigations indicate that $\mathbb{Y}$HPTM is a simple and powerful mathematical tool to deal with the complexity of such problems.

This article presents a new iterative method (NIM) for the investigation of the approximate solution of the Klein-Gordon and sine-Gordon equations. This approach is formulated on the combination of the Mohand transform and the homotopy perturbation method. Mohand transform (MT) is capable to handle the linear terms only, thus we introduce homotopy perturbation method (HPM) to tackle the nonlinear terms. This NIM derives the results in the form of a series solution. The proposed method emphasizes the stability of the derived solutions without any linearization, discretization, or hypothesis. Graphical representation and absolute error demonstrate the efficiency and authenticity of this scheme. Some numerical models are illustrated to show the compactness and reliability of this strategy.

Let G be a connected graph and d(μ,ω) be the distance between any two vertices of G. The diameter of G is denoted by diam(G) and is equal to {d(μ,ω); μ,ω ∈ G}. The radio labeling(RL) for the graph G is an injective function ϝ : V(G) → N ⋃ {0} such that for any pair of vertices μ and ω |ϝ(μ) – ϝ(ω)|≥diam(G) – d(μ,ω)+1. The span of radio labeling is the largest number in ϝ(V). The radio number of G, denoted by rn(G) is the minimum span over all radio labeling of G. In this paper, we determine radio number for the generalized Petersen graphs, P(n,2), n = 4k+2. Further the lower bound of radio number for P(n,2) when n = 4k is determined.