## 3 Results and Discussion
### Linear Fitting  
Linear regression is the most common isotherm fitting technique that uses the linear least-squares method. It provides structurally evolving isotherm equations for straightforward experimental data fitting using the coefficient of determinant. The linearization approach of the Langmuir isotherm model allows for data fitting through the six different forms of simple linear regression for various parameters. The plotting techniques of the linearized forms that are used to estimate \(q_{m}\) and \(K_{L}\) are given in Table 2. Interestingly, the linear models obtained have pairs due to the analogous iterative interexchange of variables (Table 2). The corresponding \"Type-1 & 2, Type-3 & 4, and Type-5 & 6\" result from the interchange of independent and dependent variables during the linearization of the governing equation (Figure S1) [22]. Theoretically, it is assumed that the independent variable is error-free, as it is the experimental entity. However, relocating the numerator and denominator for regression may alter the correlation results. Therefore, deviations in the different parameters of a fitted equation enable one to search for the best error distribution through comparison. All five models' linear fit isotherm parameters, including the data obtained from the six linear-transformed Langmuir models, are documented in Table 3, and the fitted curves are shown in Figure 3. Table 3 shows that the Langmuir isotherm parameters differ from each other in terms of the linear forms. Among the six Langmuir models, Type-1 and Type-2 give the highest \(R^{2}\) values for all measured temperatures [25]. Meanwhile, these two forms have a lower maximum adsorption capacity (\(q_{m}\)) than the others. These effects result from the methods adopted for extrapolating the isotherm model's linearization, where the error distribution either conforms or deflects. Figure 2: (**a**) MB adsorption capacity and dye separation efficiency. (**b**) Sorption performance comparison of the present work with other reported graphene adsorbents [56; 57; 58; 59; 60; 61; 62; 63; 64; 65]. On the other hand, the Type-4 Langmuir model form has the highest \(q_{m}\), amounting to 844.9 mg g\({}^{-1}\), with a low \(R^{2}\) value (the same as Type-3). From Table 2, we can see that at 333 K (Type-4), the highest \(q_{m}\) of 858.515 mg g\({}^{-1}\) for MB sorption was found. However, dissimilarity in the Langmuir constant (\(K_{L}\)) and a lower coefficient for particular temperatures are observed due to the equations with a structurally varying error distribution. The low \(R^{2}\) from the linear fitting of the Freundlich isotherm (Table 3) and scatter from the experimental data (Figure 3) for all temperatures show the poor fitting in the present study. Therefore, the equilibrium data obtained through MB adsorption on the AG were fitted in the linearized form of three-parameter isotherm models, namely, R-P, Sips, and Toth. Due to three unknown digits, the traditional linearization of this model was not possible. Hence, trial-and-error minimization based on computer operating techniques was applied. The R-P isotherm, after linear transformation, was plotted against \(ln\Big{(}A\frac{C_{e}}{q_{e}}-1\Big{)}\) vs. \(ln(C_{e})\). Since it is a hybrid isotherm generated by combining Langmuir and Freundlich models, each parameter, including \(A\), \(B\), and \(g\), refers to an empirical condition. Here, the exponent \(g\) signifies the nature of the adsorbent and can explain the adsorption process to an extent. Within the boundary of 0-1, if the \(g\) value tends towards 0, one presumes that the heterogeneous surface nature of the adsorbent is related to Freundlich-type adsorption and vice versa. The calculation of the unknown parameters is simplified by maximizing the coefficient \(R^{2}\) between the maximum adsorption data \(q_{e}\) and the values calculated from the linearized R-P isotherm equation.