On Upper Bounds for the Geometric-Arithmetic Topological Index

Let G = (V, E), V = {1,2,..., n}, be a simple connected graph with n >= 2 vertices, and m edges, and let Delta = d(1) >= d(2) >= center dot center dot center dot >= d(n) = delta > 0, d(i) = d(i), be a sequence of its vertex degrees. A vertex-degree-based topological index, called geometric-arithmetic index, is defined as GA = Sigma(i similar to j) 2 root d(i)d(j)/d(i)+d(j), where i similar to j denotes adjacency of vertices i and j. We first analyze some upper bounds for GA reported in the literature. The inequality GA <= m, although simple, is very important and can be used to test whether another upper bound, depending on some other parameters, has any sense. We will show that a number of upper bounds for GA reported in the literature sire worthless. Namely, if some other upper bound is greater than m, it is obviously useless. Then we determine some new upper bounds for GA in terms of some other vertex-degree-based indices.