In the present paper, we compare the geometric-arithmetic index $GA$ and the chromatic number $\chi$ of a connected graph with given order. We prove, among other results, an upper bound on the ratio $GA/\chi$. We also prove lower bounds on the chromatic number in terms of geometric-arithmetic index and number of vertices of a connected graph. The results obtained for the chromatic number $\chi$ are extended to the clique number $\omega$.