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Polynomials with an exponential equivalent to the extent of a characteristic partition and coefficients proportional to the multiplicity/occurrence of the accompanying partition are termed as counting polynomials. These polynomials are a well-known method for describing a chemical graph's molecular invariants as polynomials. It is possible to deduce several key topological invariants from polynomials by either directly taking their value at a certain point or by calculating derivatives or integrals of the polynomial. A topological invariants is a real number correlated with a network that predicts the physico-chemical properties. Chemical modeling, drug design, and structural activity relations use invariants. This paper aims to find the counting polynomials such as Sadhana, omega, PI and theta polynomial of certain graphene nanostructures. In furthermore, topological invariants of specific graphene nanostructures that are related to these counting polynomials are investigated. [ABSTRACT FROM AUTHOR]

Networks play an important role in electrical and electronic engineering. It depends on what area of electrical and electronic engineering, for example there is a lot more abstract mathematics in communication theory and signal processing and networking etc. Networks involve nodes communicating with each other. Graph theory has found a considerable use in this area of research. A topological index is a real number associated with chemical constitution purporting for correlation of chemical networks with various physical properties, chemical reactivity. The concept of hyper Zagreb index, first multiple Zagreb index, second multiple Zagreb index and Zagreb polynomials was established in chemical graph theory based on vertex degrees. In this paper, we extend this study to interconnection networks and derive analytical closed results of hyper Zagreb index, first multiple Zagreb index, second multiple Zagreb index, Zagreb polynomials and redefined Zagreb indices for block shift network (BSN − 1) and (BSN − 2), hierarchical hypercube (HHC − 1) and (HHC − 2).

In this study, a nonlinear mathematical SIR system is explored numerically based on the dynamics of the waterborne disease, e.g., cholera, that is used to incorporate the delay factor through the antiseptics for disease control. The nonlinear mathematical SIR system is divided into five dynamics, susceptible X(u), infective Y(u), recovered Z(u) along with the B(u) and Ch(u) be the contaminated water density. Three cases of the SIR system are observed using the artificial neural network (ANN) along with the computational Levenberg-Marquardt backpropagation (LMB) called ANNLMB. The statistical performances of the SIR model are provided by the selection of the data as 74% for authentication and 13% for both training and testing, together with 12 numbers of neurons. The exactness of the designed ANNLMB procedure is pragmatic through the comparison procedures of the proposed and reference results based on the Adam method. The substantiation, constancy, reliability, precision, and ability of the proposed ANNLMB technique are observed based on the state transitions measures, error histograms, regression, correlation performances, and mean square error values. [ABSTRACT FROM AUTHOR]

In this paper, the rational sine–cosine and rational sinh–cosh methods are applied in extracting some properties of nonlinear Phi-four and Gross–Pitaevskii equations. The singular periodic wave solutions, dark soliton solutions and hyperbolic function solutions are reported. The solitary waves are observed from the traveling waves under the values of the parameters. Modulation instability analysis is also observed in various simulations. We also plot to observe the wave distributions of parameters of stability in 2D and 3D visuals via package program. [ABSTRACT FROM AUTHOR]