@article {
author = {Lim, Johnny and Chew, Zheng Kiat and Lim, Macco Zhi Pei and Thoo, Kai Jie},
title = {Quantization of Sombor Energy for Complete Graphs with Self-Loops of Large Size},
journal = {Iranian Journal of Mathematical Chemistry},
volume = {14},
number = {4},
pages = {225-241},
year = {2023},
publisher = {University of Kashan},
issn = {2228-6489},
eissn = {2008-9015},
doi = {10.22052/ijmc.2023.252770.1707},
abstract = {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.},
keywords = {Euclidean metric,Sombor energy,Sombor spectrum,Graphs with self-loops},
url = {https://ijmc.kashanu.ac.ir/article_114116.html},
eprint = {https://ijmc.kashanu.ac.ir/article_114116_27abc540d655033b358779bc2dd1c63d.pdf}
}