A complete solution for maximizing the general Sombor index of chemical trees with given number of pendant vertices

For a graph G, the general Sombor (SOα) index is defined as:SOα(G)=∑vivj∈E(G)(di2+dj2)α, where α≠0 is a real number, E(G) is the edge set and di denotes the degree of a vertex vi in G. A chemical tree is a tree in which no vertex has a degree greater than 4, and a pendant vertex is a vertex with degree 1. This paper aims to completely characterize the n− vertex chemical trees with a fixed number of pendant vertices (=p) that maximize the SOα index over α0<α<α1, where α0≈0.144 is the unique non-zero root of equation 4(32α−25α)+8α−13α+5α−10α=0 and α1≈3.335 is the unique non-zero solution of equation 3(17α−10α)+3(20)α−13α−2(25)α=0. Since SO1 and SO12 correspond to the classical forgotten and the Sombor indices of a graph G, respectively, our results apply to both indices. Moreover, Liu et al. [More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons, Int. J. Quantum Chem. 121 (2021) #26689] addressed the problem of maximizing the Sombor index for chemical trees with even p≥6 only, which was later extended by Du et al. [On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves, Appl. Math. Comput. 464 (2024) #128390] to include both even p≥6 and odd p≥9. This paper, in contrast, provides a more comprehensive solution, fully characterizing the problem for all p≥3 maximizing the general Sombor index for any α, where α0<α<α1. In addition, the chemical significance of the SOα index over the range −10≤α≤10 is explored by using the octane isomers dataset to predict their physicochemical properties. Promising results are obtained when the approximated values of α belong to the set {−1,1,8,10}.