Spectral analysis of Cupric oxide (CuO) and Graphene Oxide (GO) via machine learning techniques

Chemical graph theory has recently gained much attraction among researchers due to its extensive use in mathematical chemistry. In this research article, We have studied the spectral properties such as eigenvalues, energy and Estrada index of some chemical structures such as Cupric oxide ($CuO$) and Graphene Oxide (GO). We have computed the energy $E\left(G\right)={\sum }_{i=0}^n\left|{\lambda }_i\right|$ and the other invariant Estrada index $EE\left(G\right)={\sum }_{i=1}^n{e}^{{\lambda }_i}$ of the above mentioned graph structures and obtain the polynomial regression analysis using machine learning techniques. This approach permitted us to predict the spectral values more precisely and analyze the difference between the actual and predicted values. The actual values of energy and Estrada index is represented by $E_{av}$ and $E{E}_{av}$ while the predicted values of energy and Estrada index is represented by $E_{pv}$ and $E{E}_{pv}$, where $av$ represents ”actual value” and $pv$ represents ”predicted value”. We first use traditional method based on softwares and get the actual values ($av$) (see section 2). Then we perform machine learning techniques to generate a best fit model and get the predicted values ($pv$) of the energies and Estrada index of Cupric oxide $CuO$ and Graphene Oxide $GO$ by using the best fit second order polynomial for Energy and Estrada Index of $CuO$ is obtained as $E\left(\mathrm{CuO}\right)=-0.007m^2+5.892mn+2.243m+2.169n-0.365$ and $EE\left(\mathrm{CuO}\right)=0.537+2.084m+2.084n+13.533mn$, respectively. Similarly, the best fit second order polynomial for Energy and Estrada Index of $GO$ is obtained as $E\left(\mathrm{GO}\right)=-0.266+2.533m+2.598n+0.014m^2+3.133mn+-0.017n^2$ and $EE\left(\mathrm{GO}\right)=-1.671+4.400m+4.440n+0.016m^2+6.553mn+0.014n^2$, respectively. We have observed the difference between $av$ and $pv$ which shows our machine learning model is best fit model as the error between the $av$ and $pv$ is minimum.