Quantization of Sombor Energy for ‎Complete ‎Graphs with‎ ‎Self-Loops of‎ ‎Large Size

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.