Characterization of graphs having extremal Randić indices

The higher Randić index R_t(G) of a simple graph G is defined asR_t (G) = under(∑, i₁ i₂ ⋯ i_{t + 1}) frac(1, sqrt(δ_i1 δ_i2 ⋯ δ_{i t +1})), where δ_i denotes the degree of the vertex i and i₁i₂⋯i_t+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.