First Zagreb energy of self-looped graphs: predictive insights into kidney infection drugs and theoretical bounds

Graphs containing self-loops provide a versatile framework for modeling heteroatomic molecules, with each self-loop representing a hetero-atom. In this study, we investigate the predictive capability of the first Zagreb energy in relation to the physicochemical properties of kidney infection drugs, using their corresponding molecular graphs with self-loops. The analysis using linear, quadratic, cubic, and logarithmic regression models reveals a strong correlation between the first Zagreb energy and key physicochemical properties such as polarizability, molar refractivity, and molar volume. Statistical metrics such as standard error (SE), F-test value, standard error of fit (SF), and root mean squared error (RMSE) validate the stability and reliability of the proposed models. Furthermore, we compute the first Zagreb energy of the complete graph (K₍) ₒ, as well as the complete bipartite graph (K₌, ₍) ₒ, with partite sets M=S, and N. In addition, we derive both lower and upper bounds for the first Zagreb energy of graphs with self-loops.