A spectral approach to the Randić index

The higher Randić index Rt of a simple graph Γ is defined asRt=∑vi1-vi2-⋯-vit+11δi1δi2⋯δit+1,where δi denotes the degree of the vertex vi and vi1-vi2-⋯-vit+1 runs over all paths of length t in Γ. In this paper we introduce a suitable version of the Laplacian of a graph and we formulated R1 in terms of its spectrum. Moreover, bounds on R2 from the eigenvalues either the adjacency matrix or the Laplacian matrix of the graph are obtained in the paper.