Characterization of graphs having extremal Randić indices (Q865454)

The higher Randić index \(R_t(G)\) of a simple graph \(G\) is defined as \[ R_t(G) = \sum_{i_1i_2\dots i_{t+1}}\frac{1}{\sqrt{\delta_{i_1}\delta_{i_2} \dots\delta_{i_{t+1}}}}\,, \] where \(\delta_i\) denotes the degree of the vertex \(i\) and \(i_1i_2\dots i_{t+1}\) runs over all paths of length \(t\) in \(G\). In [\textit{J. A. Rodríguez}, Linear Algebra Appl. 400, 339-344 (2005; Zbl 1063.05097)] the lower and upper bound on \(R_1(G)\) were done in terms of kinds of adjacency and Laplacian spectra. In this work the authors characterize the graphs which achieve the upper or lower bound of \(R_1(G)\) and \(R_2(G)\), respectively.