a note on the additively weighted mostar index of graphs

‎in this article‎, ‎we discuss the additively weighted mostar index‎, ‎an innovative topological measure that extends the traditional mostar index by incorporating edge weights computed as the sum of the degrees of their end point vertices‎. ‎we focus on its application within the set mathcal{u}_n comprising all unicyclic graphs of order n‎. ‎our study rigorously establishes the first two sharp lower and upper bounds for this index across graphs in mathcal{u}_n‎. ‎additionally‎, ‎we analyze the additively weighted mostar index of cartesian product graphs and investigate its properties across various graph classes‎. ‎furthermore‎, ‎we demonstrate the practical utility of this index by comparing its effectiveness against eight other distance-based topological indices in predicting chemical properties of octane isomers‎. ‎remarkably‎, ‎we find that the additively weighted mostar index outperforms or matches the predictive capabilities of other indices in linear models with these chemical properties‎, ‎highlighting its potential in quantitative structure-property relationships‎. ‎this research significantly contributes to both graph theory and chemical informatics by showcasing the unique advantages of the additively weighted mostar index in structural analysis and predictive modeling‎.