Extremal graphs for normalized Laplacian spectral radius and energy

Let G = (V, E) be a simple graph of order n and the normalized Laplacian eigenvalues ρ₁ ≥ ρ₂ ≥ ··· ≥ ρ_n−1 ≥ ρ_n = 0. The normalized Laplacian energy (or Randić energy) of G without any isolated vertex is defined as (Formula Present). In this paper, a lower bound on ρ₁ of connected graph G (G is not isomorphic to complete graph) is given and the extremal graphs (that is, the second minimal normalized Laplacian spectral radius of connected graphs) are characterized. Moreover, Nordhaus-Gaddum type results for ρ₁ are obtained. Recently, Gutman et al. gave a conjecture on Randić energy of connected graph [I. Gutman, B. Furtula, Ş. B. Bozkurt, On Randić energy, Linear Algebra Appl. 442 (2014) 50-57]. Here this conjecture for starlike trees is proven.