Characterization of graphs having extremal Randić indices

The higher Randić index Rt(G) of a simple graph G is defined as where δi denotes the degree of the vertex i and i1i2 it+1 runs over all paths of length t in G. In [J.A. Rodríguez, A spectral approach to the Randić index, Linear Algebra Appl. 400 (2005) 339–344], the lower and upper bound on R1(G) was determined in terms of a kind of Laplacian spectra, and the lower and upper bound on R2(G) were done in terms of kinds of adjacency and Laplacian spectra. In this paper we characterize the graphs which achieve the upper or lower bounds of R1(G) and R2(G), respectively.