TY - JOUR
ID - 114116
TI - Quantization of Sombor Energy for Complete Graphs with Self-Loops of Large Size
JO - Iranian Journal of Mathematical Chemistry
JA - IJMC
LA - en
SN - 2228-6489
AU - Lim, Johnny
AU - Chew, Zheng Kiat
AU - Lim, Macco Zhi Pei
AU - Thoo, Kai Jie
AD - School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia
Y1 - 2023
PY - 2023
VL - 14
IS - 4
SP - 225
EP - 241
KW - Euclidean metric
KW - Sombor energy
KW - Sombor spectrum
KW - Graphs with self-loops
DO - 10.22052/ijmc.2023.252770.1707
N2 - 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.
UR - https://ijmc.kashanu.ac.ir/article_114116.html
L1 - https://ijmc.kashanu.ac.ir/article_114116_27abc540d655033b358779bc2dd1c63d.pdf
ER -