The geometric-arithmetic index and the chromatic number of connected graphs

In the present paper, we compare the geometric-arithmetic index GA and the chromatic number χ of a connected graph with given order. We prove, among other results, an upper bound on the ratio GA∕χ. 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 χ are extended to the clique number ω.