Improved bounds for the largest eigenvalue of trees

Let T be a tree with vertex set V. Let dv denotes the degree of v ∈ V. Let Δ = max{dv : v ∈ V}. Let u ∈ V such that du = Δ. Let k = eu + 1 where eu is the excentricity of u. For j = 1, 2, …, k − 2, letδj=maxdv:distv,u=j.We prove thatμ1T