Characterization of graphs having extremal Randic indices

Abstract

The higher Randic index R-t(G) of a simple graph G is defined as [GRAPHICS] where delta(i) denotes the degree of the vertex i and i(1)i(2) ... i(t+1) runs, over all paths of length t in G. In [J.A. Rodriguez, A spectral approach to the Randic index, Linear Algebra Appl. 400 (2005) 339-344], the lower and upper bound on R-1(G) was determined in terms of a kind of Laplacian spectra, and the lower and upper bound on R-2(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 R-1(G) and R-2(G), respectively. (c) 2006 Elsevier Inc. All rights reserved.