Normalized Laplacian Eigenvalues and Energy of Trees

Let G be a graph with vertex set V(G) = {v(1), v(2),...,v(n)} and edge set E(G). For any vertex v(i) epsilon V(G), let d(i) denote the degree of v(i). The normalized Laplacian matrix of the graph G is the matrix L = (L-ij) given by [GRAPHICS] In this paper, we obtain some bounds on the second smallest normalized Laplacian eigenvalue of tree T in terms of graph parameters and characterize the extremal trees. Utilizing these results we present some lower bounds on the normalized Laplacian energy (or Randie energy) of tree T and characterize trees for which the bound is attained.