Remark on spectral study of the geometric-arithmetic index and some generalizations

Let G = (V, E), V = {1, 2,..., n}, be a simple connected graph with n vertices, m edges, and sequence of vertex degrees d(1) >= d(2) >= ... >= d(n) > 0, d(i) = d(i). A large number of vertex-degree-based topological indices is of the form TI = I(G) = Sigma(i similar to j) F (d(i), d(j)), where F is pertinently chosen function with the property F (x, y) = F (y, x). To each of such topological indices a corresponding adjacency matrix A = (a(ij)), of order n x n, can be associated. The trace of matrix A is denoted as tr(A). For F (d(i), d(j)) = 2 root d(i)d(i)/d(i)+d(i), the geometric-arithmetic topological index, GA(1), is obtained. Upper and lower bounds for GA(1) in terms of tr(A(2) ) are determined. Also, we generalize a number of results reported in the literature and obtain some new bounds for the indices of the form TI. (c) (C) 2018 Elsevier Inc. All rights reserved.