New upper bounds on the spectral radius of trees with the given number of vertices and maximum degree

This paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree. We obtain the new upper bounds on the spectral radius of the trees, and the results are the best upper bounds expressed by the number of vertices and maximum degree, at present.Let T=(V,E) be a tree on n vertices with maximum degree Δ, where 3⩽Δ⩽n−2. Denote by ρ(T) the spectral radius of T. We prove that(1)if n⩽2Δ, then ρ(T)⩽n−1+(n−2Δ)2+2n−32, and equality holds if and only if T is an almost completely full-degree tree of 3 levels;(2)if 2ΔΔ2+1, then ρ(T)<2Δ−1cosπ2k+1, where k=⌈logΔ−1((Δ−2)(n−1)Δ+1)⌉+1.