%0 Journal Article
%T Quantization of Sombor Energy for Complete Graphs with Self-Loops of Large Size
%J Iranian Journal of Mathematical Chemistry
%I University of Kashan
%Z 2228-6489
%A Lim, Johnny
%A Chew, Zheng Kiat
%A Lim, Macco Zhi Pei
%A Thoo, Kai Jie
%D 2023
%\ 12/01/2023
%V 14
%N 4
%P 225-241
%! Quantization of Sombor Energy for Complete Graphs with Self-Loops of Large Size
%K Euclidean metric
%K Sombor energy
%K Sombor spectrum
%K Graphs with self-loops
%R 10.22052/ijmc.2023.252770.1707
%X 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 $\sigma$.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.
%U https://ijmc.kashanu.ac.ir/article_114116_27abc540d655033b358779bc2dd1c63d.pdf