Remarks on Bounds of Normalized Laplacian Eigenvalues of Graphs

Let $G$ be a connected undirected graph with $n$, $nge 3$, vertices and $m$ edges. Denote by $rho_1 ge rho_2 ge cdots > rho_n =0$ the normalized Laplacian eigenvalues of $G$. Upper and lower bounds of $rho_i$, $i=1,2,ldots , n-1$, are determined in terms of $n$ and general Randi c index, $R_{-1}$.