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

A self-loop graph GS is a simple graph G obtained by attaching loops at S V (G): To such GS 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 GS; for which a Forgotten-like index arises. We explicitly study the Sombor energy ESO of complete graphs with self-loops (Kn)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 is then studied. Surprisingly, the constant 4 p 2 is obtained repeatedly in several scenarios, yielding a quantization of the energy change of 1 loop for large n and . Finally, we provide a McClelland-type and determinantal-type upper and lower bounds for ESO(GS); which generalizes several bounds in the literature.