Enhanced Mathematical Models for the Sombor Index: Reduced and Co-Sombor Index Perspectives

In this study, we extend the mathematical framework of the Sombor index and its variants by developing enhanced versions: The Enhanced Sombor Index (ESO), Enhanced Reduced Sombor Index (ERSO), and Enhanced co-Sombor Index (ECSO). These enhanced indices incorporate weighting functions to provide a more nuanced analysis of graph properties. We derive key properties and theorems, demonstrating that the enhanced indices are at least as large as their traditional counterparts. We also establish upper bounds for these indices in bipartite graphs, specifically K₃, 3. Practical applications in chemical graph theory, social network analysis, and biological networks illustrate the utility of these enhanced indices. Detailed calculations for the complete bipartite graph K₃, 3 validate our theoretical findings and demonstrate the practical computation of the indices. Potential future research directions include generalization to other graph classes, optimization of weighting functions, algorithmic development, application to dynamic networks, empirical validation, and interdisciplinary applications.