A Unified Majorization Approach to Extremal General Zeroth-Order Randić Indices

The general zeroth-order Randić index Rα0(G)=∑v∈Vd(v)α (α≠0,1) unifies several classical indices, including the first Zagreb (α=2), forgotten (α=3), and inverse degree (α=−1) indices. We characterize the connected graphs that attain extremal Rα0 values for most parameter regimes. Nested stars maximize for α≥2, quasi-stars for α<0, and quasi-regular graphs for 0<α<1. For 1<α<2, we show that the maximizer is a threshold graph, but the exact structure depends on n, m, and α; a counterexample demonstrates that the previously claimed optimality of the quasi-star in this regime is not always valid, and a complete characterization remains open. Minimizers follow the opposite pattern. Our majorization-based framework, combined with the sign of the third derivative of xα, yields explicit closed-form formulas for the extremal values. We also extend the analysis to disconnected graphs under δ(G)≥1, identifying star-forests and clique-forests as the extremal maximizers (for α≥2 and α<0, respectively), while the regimes 0<α<1 and 1<α<2 are addressed with quasi-regular graphs and left as open problems, respectively.