Abstract

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. The α−index of G is the spectral radius ρα (G) of the matrix Aα (G) = αD (G) + (1 −α)A (G), where α ∈ [0, 1]. Let Tn,k be the tree of order n and k pendent vertices obtained from a star K1,k and k pendent paths of almost equal lengths attached to different pendent vertices of K1,k. It is shown that if α ∈ [0, 1) and T is a tree of order n with k pendent vertices then ρα(T) ≤ ρα(Tn,k), with equality holding if and only if T = Tn,k. This result generalizes a theorem of Wu, Xiao and Hong [6] in which the result is proved for the adjacency matrix (α = 0). Let q = [n−1k ] and n − 1 = kq + r, 0 ≤ r ≤ k − 1. It is also obtained that the spectrum of Aα(Tn,k) is the union of the spectra of two special symmetric tridiagonal matrices of order q and q + 1 when r = 0 or the union of the spectra of three special symmetric tridiagonal matrices of order q, q + 1 and 2q + 2 when r ≠ 0. Thus, the α−index of Tn,k can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order q + 1 if r = 0 or order 2q + 2 if r ≠ 0.