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

Abstract

Let G=(V,E),V={1,2,…,n}, be a simple connected graph with n vertices, m edges, and sequence of vertex degrees d1 ≥ d2 ≥ ⋅⋅⋅ ≥ dn > 0, di=d(i). A large number of vertex–degree-based topological indices is of the form TI=TI(G)=∑i∼jF(di,dj), 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=(aij), of order n × n, can be associated. The trace of matrix A is denoted as tr(A). For F(di,dj)=2didjdi+dj, the geometric–arithmetic topological index, GA1, is obtained. Upper and lower bounds for GA1 in terms of tr(A2) 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.