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.