Estimating the higher-order Randić index

Abstract

Let G be a (molecular) graph with vertex set V = {v1, v2, ..., vn}. Let δ (vi) be the degree of the vertex vi ∈ V. If the vertices vi1, vi2, ..., vih + 1 form a path of length h, h ≥ 1, in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1 / sqrt(δ (vi1) δ (vi2), ..., δ (vih + 1)) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for Rh are obtained, in terms of the vertex degree sequence of G. Closed formulas for Rh are obtained for the case when G is regular or semiregular bipartite. © 2010 Elsevier B.V. All rights reserved.