Comparing the geometric–arithmetic index and the spectral radius of graphs

The geometric-arithmetic index GA of a graph G is the sum of ratios, over all edges of G, of the geometric mean to the arithmetic mean of the end vertices degrees of an edge. The spectral radius λ₁ of G is the largest eigenvalue of its adjacency matrix. These two parameters are known to be used as molecular descriptors in chemical graph theory. In the present paper, we compare GA and λ₁ of a connected graph with given order. We prove, among other results, upper and lower bounds on the ratio GA/λ1 as well as a lower bound on the ratio GA/λ²₁. In addition, we characterize all extremal graphs corresponding to each of these bounds.