Estimating the higher-order Randić index

Let G be a (molecular) graph with vertex set V = { v 1 , v 2 , … , v n } . Let δ ( v i ) be the degree of the vertex v i ∈ V . If the vertices v i 1 , v i 2 , … , v i h + 1 form a path of length h , h ⩾ 1 , in the graph G, then the hth order Randić index R h of G is defined as the sum of the terms 1 / δ ( v i 1 ) δ ( v i 2 ) , … , δ ( v i h + 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.