On the difference between Sombor index and harmonic-arithmetic index and its applications

The Sombor index ($ SO $) and harmonic-arithmetic index ($ HA $) stand as two well-established topological indices constructed based on vertex degrees. The difference between the $ SO $ index and $ HA $ index of a graph G is introduced as(SO-HA)(G) = ∑_uv∈ E(G)√d(u)²+d(v)²-∑_uv∈ E(G)4d(u)d(v)/(d(u)+d(v))²,where $ d(u) $ stands for the degree of vertex u∈ V(G). In the present work, we determined the trees, unicyclic graphs, and bicyclic graphs that achieved extremal $ SO-HA $ values, characterized the chemical graphs with extremal $ SO-HA $ values, and obtained the trees with fixed maximum degree minimizing the $ SO-HA $ index. Furthermore, by performing QSPR analysis on octane isomers, we observed that the $ SO-HA $ index possessed favorable physicochemical predictability.