A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider $\mathcal{G}$ as a group and H as a non-empty subset of $\mathcal{G}$. The commuting graph $\mathcal{C}(\mathcal{G},H)$, has H as the vertex set, where $h_1,h_2\in H$ are edge connected whenever $h_1$ and $h_2$ commute in $\mathcal{G}$. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of $SL(2,\mathbb{C})$. Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups.

We utilize Lyapunov exponents to quantitatively assess the hyperchaos and categorize the limit sets of complex dynamical systems. While there are numerous methods for computing Lyapunov exponents in integer-order systems, these methods are not suitable for fractional-order systems because of the nonlocal characteristics of fractional-order derivatives. This paper introduces innovative eight-dimensional chaotic systems that investigate fractional-order dynamics. These systems exploit the memory effect inherent in the Grünwald–Letnikov (G-L) derivative. This approach enhances the system’s applicability and compatibility with traditional integer-order systems. An 8D Chen’s fractional-order system is utilized to showcase the effectiveness of the presented methodology for hyperchaotic systems. The simulation results demonstrate that the proposed algorithm outperforms existing algorithms in both accuracy and precision. Moreover, the study utilizes the 0–1 Test for Chaos, Kolmogorov–Sinai (KS) entropy, the Kaplan–Yorke dimension, and the Perron Effect to analyze the proposed eight-dimensional fractional-order system. These additional metrics offer a thorough insight into the system’s chaotic behavior and stability characteristics.

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph ${\mathrm{\Gamma }}_{\mathcal{G}}$ of a finite group $\mathcal{G}$ is a graph where non-central elements of $\mathcal{G}$ are its vertex set, while two different elements are edge connected when they do not commute in $\mathcal{G}$. In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of $SL(2,\mathbb{C})$. We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.

In this research paper, we deal with the problem of determining the function $\chi :G\to \mathbb{R}$, which is the solution to the maximum functional equation (MFE) $max\{\chi \left(xy\right),\chi \left(x{y}^{-1}\right)\}=\chi \left(x\right)\chi \left(y\right),$ when the domain is a discretely normed abelian group or any arbitrary group G. We also analyse the stability of the maximum functional equation $max\{\chi \left(xy\right),\chi \left(x{y}^{-1}\right)\}=\chi \left(x\right)+\chi \left(y\right)$ and its solutions for the function $\chi :G\to \mathbb{R}$, where G be any group and also investigate the connection of the stability with commutators and free abelian group K that can be embedded into a group G.

In the current investigation, we propose the analytical and numerical solutions of the Navier–Stokes equation for the reverse roll coating process (RRCP) using a bath of the Sisko fluid model (SFM). A suitable transformation is applied to the partial differential equation-based mathematical model of SFM-RRCP, resulting in a set of nonlinear ordinary differential equations. The perturbation method has been employed to find the analytical solutions for velocity, flow rate, pressure gradient, pressure, and temperature distribution. The Newton–Raphson method has been used to find coating thickness. Furthermore, numerical integration has been applied to compute some mechanical quantities of interest, including power input and roll separation force. Sensitivity analysis is another approach implemented using response surface methods to examine the impacts of velocity ratio and non-Newtonian parameters on coating thickness, power input, and roll separating forces. The residual graphs and contour diagrams are also shown. It has been observed that as the value of non-Newtonian fluid parameters increases, the velocity profile decreases. However, the coating thickness on the web is a decreasing function. Further, it has been observed that separation points moved toward the nip region when the value of the velocity ratio was raised. Moreover, the sensitivity of roll separating forces and power input to input parameters is negative.