In this research article, we replace the integer degree \( n=\deg(v) \) of a vertex by its \( q(t) \)-analogue \( [n]_{q(t)} \) and by an \( h(t) \)-analogue \( \langle n\rangle_{h(t)} \) obtained from the \( h(t) \)-derivative of \( x^{n} \) at \( x=1 \), and we show that every classical degree-based index \( \mathrm{TI}(G)=\sum_{uv\in E(G)}F(\deg u,\deg v) \) admits two functional companions \( \mathrm{TI}_{q(t)}(G) \) and \( \mathrm{TI}_{h(t)}(G) \) that recover \( \mathrm{TI}(G) \) exactly in the limit \( t\to 0 \).
We establish convergence, monotonicity, boundedness and a functional congruence theorem for these deformed indices, specialise the construction to the first and second Zagreb indices, the Randić index, the ABC index, the GA index, the harmonic index and the Sombor index, and verify all theoretical claims numerically on path, cycle, star and fused-ring (naphthalene) molecular graphs.
We then give an explicit chemical application, showing how an edge-weighted extension of the same framework tracks a degree-based topological index continuously along the reaction coordinate of a Diels-Alder cycloaddition, recovering the classical reactant- and product-graph indices at the two endpoints.