A Novel Forgotten-Sombor Energy: Bounds and Applications to Anti-Diabetes Drugs

Abstract

In this paper, we introduce a novel matrix naming it to be Forgotten-Sombor matrix, denoted by ℱ????????(????), for a simple graph ????. This matrix is defined such that for vertices ???????? adjacent to ???????? , the (????,????)-entry is given by √????????(????????) 4 + ????????(????????) 4, where ????????(????????) represents the degree of the vertex ???????? , and zero otherwise. Let ????1 ≥ ????2 ≥ ⋯ ≥ ???????? denote the eigenvalues of ℱ????????(????), with ????1 corresponding to the spectral radius, and let the associated Forgotten-Sombor energy ????ℱ????????(????) be defined as the sum of the absolute values of these eigenvalues. We derived new upper and lower bounds for both ????1 and ????ℱ????????(????) in terms of the first Zagreb index ????1(????). As an application, we conduct a Quantitative Structure-Property Relationship (QSPR) analysis on a dataset comprising drugs used in the treatment of Diabetese, constructing linear regression models to explore the correlation between the physicochemical properties of these drugs and their corresponding ????ℱ????????(????) values, with results illustrated through graphical representations.