University of KashanIranian Journal of Mathematical Chemistry2228-648914420231201Quantization of Sombor Energy for Complete Graphs with Self-Loops of Large Size22524111411610.22052/ijmc.2023.252770.1707ENJohnnyLimSchool of Mathematical Sciences, Universiti Sains Malaysia, Malaysia0000-0002-4562-1869Zheng KiatChewSchool of Mathematical Sciences, Universiti Sains Malaysia, Malaysia0009-0009-9890-7391Macco Zhi PeiLimSchool of Mathematical Sciences, Universiti Sains Malaysia, Malaysia0009-0009-7088-9013Kai JieThooSchool of Mathematical Sciences, Universiti Sains Malaysia, Malaysia0009-0008-7171-5149Journal Article20230408A self-loop graph $G_S$ is a simple graph $G$ obtained by attaching loops at $S \subseteq V(G).$ To such $G_S$ an Euclidean metric function is assigned to its vertices, forming the so-called Sombor matrix. In this paper, we derive two summation formulas for the spectrum of the Sombor matrix associated with $G_S,$ for which a Forgotten-like index arises. We explicitly study the Sombor energy $\cE_{SO}$ of complete graphs with self-loops $(K_n)_S,$ as the sum of the absolute value of the difference of its Sombor eigenvalues and an averaged trace. The behavior of this energy and its change for a large number of vertices $n$ and loops $\sigma$ is then studied. Surprisingly, the constant $4\sqrt{2}$ is obtained repeatedly in several scenarios, yielding a quantization of the energy change of 1 loop for large $n$ and $\sigma$.<br />Finally, we provide a McClelland-type and determinantal-type upper and lower bounds for $\cE_{SO}(G_S),$ which generalizes several bounds in the literature.https://ijmc.kashanu.ac.ir/article_114116_27abc540d655033b358779bc2dd1c63d.pdf