University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 14 4 2023 12 01 Quantization of Sombor Energy for ‎Complete ‎Graphs with‎ ‎Self-Loops of‎ ‎Large Size 225 241 114116 10.22052/ijmc.2023.252770.1707 EN Johnny Lim School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia Zheng Kiat Chew School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia Macco Zhi Pei Lim School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia Kai Jie Thoo School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia Journal Article 2023 04 08 ‎A 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