Normalized Laplacian eigenvalues and energy of trees

Let G be a graph with vertex set V (G) = {v₁,v₂,…,v_n} and edge set E(G). For any vertex v_i ∈ 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 (Formula Presented) 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 Randić energy) of tree T and characterize trees for which the bound is attained.