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More on Comparison Between First Geometric-Arithmetic Index and Atom-Bond Connectivity Index

The first geometric-arithmetic (GA) index and atom-bond connectivity (ABC) index are molecular structure descriptors which play a significant role in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies. Das and Trinajstić [\textit{Chem. Phys. Lett.} \textbf{497} (2010) 149-151] showed that $GA$ index is greater than $ABC$ index for all those graphs (except $K_{1,4}$ and $T^{*}$, see Figure 1) in which the difference between maximum and minimum degree is less than or equal to 3. In this note, it is proved that $GA$ index is greater than $ABC$ index for line graphs of molecular graphs, for general graphs in which the difference between maximum and minimum degree is less than or equal to $(2δ-1)^{2}$ (where $δ$ is the minimum degree and $δ\geq2$) and for some families of trees. Thereby, a partial solution to an open problem proposed by Das and Trinajstić is given.