Estimating the higher-order Randic index

Let G be a (molecular) graph with vertex set V = {v(1), v(2), ..., v(n)}. Let delta(v(i)) be the degree of the vertex v(i) is an element of V. If the vertices v(i1), v(i2), ... , v(ih+1) form a path of length h, h >= 1, in the graph G, then the hth order Randic index R-h of G is defined as the sum of the terms 1/root delta(v(i1))delta(v(i2)), ... , delta(v(ih+1)) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for R-h are obtained, in terms of the vertex degree sequence of G. Closed formulas for R-h are obtained for the case when G is regular or semiregular bipartite. (C) 2010 Elsevier B.V. All rights reserved.