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The first geometric-arithmetic (GA) index and atom-bond connectivity (ABC)index are molecular structure descriptors which play a significant role inquantitative structure-property relationship (QSPR) and quantitativestructure-activity relationship (QSAR) studies. Das and Trinajsti\'{c}[\textit{Chem. Phys. Lett.} \textbf{497} (2010) 149-151] showed that $GA$ indexis 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 isless than or equal to 3. In this note, it is proved that $GA$ index is greaterthan $ABC$ index for line graphs of molecular graphs, for general graphs inwhich the difference between maximum and minimum degree is less than or equalto $(2\delta-1)^{2}$ (where $\delta$ is the minimum degree and $\delta\geq2$)and for some families of trees. Thereby, a partial solution to an open problemproposed by Das and Trinajsti\'{c} is given.

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