Geometric-arithmetic index and minimum degree of connected graphs

In the present paper, we prove lower and upper bounds for each of the ratios \(GA/\delta\), as well as a lower bound on \(GA/\sqrt{\delta}\), in terms of the order \(n\), over the class of connected graphs on \(n\) vertices, where \(GA\) and \(\delta\) denote the geometric-arithmetic index and the minimum degree, respectively. We also characterize the extremal graphs corresponding to each of those bounds. In order to prove our results, we provide a modified statement of a well-known lower bound on the geometric-arithmetic index in terms of minimum degree.