For a graph G, the total irregularity index \(irr_t(G)\) is defined as: \(\begin{aligned} irr_t(G) = \sum _{\begin{array}{c} \{v_i, v_j\} \subseteq V_G \\ i \ne j \end{array}} |d_i - d_j|, \end{aligned}\) where \( d_i \) signifies the degree of vertex \( v_i \in V_G \) . Recently, Ali et al. [Bounds and optimal results for the total irregularity measure, MATCH Commun. Math. Comput. Chem. 94 (2025) 5–29] posed an open problem regarding the characterization of molecular trees that maximize the total irregularity index \( irr_t \) among all \( n \) -vertex molecular trees with a fixed number of segments or branching vertices. In this paper, we present a complete solution to this open problem by identifying the structural properties of molecular trees that achieve the maximum \( irr_t \) . Our results not only provide explicit constructions of such optimal trees but also establish bounds and insights into the interplay between the total irregularity index, the number of segments, and branching vertices. These findings contribute to a deeper understanding of the combinatorial properties of molecular trees and their applications in mathematical chemistry.
