Structural Exploration and Expected values of Topological Indices of Square-Hexagonal <i>T</i> _2,3 Kink Chains for Advanced Engineering Applications

Square-hexagonal kink chains offer a compelling balance between geometric regularity and topological complexity, making them valuable models for studying molecular diversity in chemical graph theory. These structures, composed of alternating square and hexagonal polygons, introduce angular distortions and connectivity variations that significantly influence properties such as electronic behavior, entropy, and mechanical stability, characteristics relevant to graphene derivatives and polycyclic hydrocarbons. This work focuses on kink chains of type <i>T</i> _2,3, for which we construct four structural configurations by systematic polygonal attachments. For each configuration, we derive exact expressions for several topological indices, including geometric-arithmetic, Zagreb (first and second), Forgotten, sum-connectivity, product-connectivity, and atom bond-connectivity indices. In addition to deterministic values, we calculate exact expressions of <i>expected values</i> using a probabilistic vertex-weighting approach for these indices, thereby offering a statistical dimension to the structural analysis. Numerical evaluation, graphical visualization, and analytical comparison reveal trends and extremal indices across configurations. These findings deepen our understanding of how kink topology influences both exact and expected topological descriptors, supporting future applications in nanomaterial design and theoretical modeling.