THE MAXIMAL α-INDEX OF TREES WITH K PENDENT VERTICES AND ITS COMPUTATION

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. The alpha-index of G is the spectral radius rho(alpha) (G) of the matrix A(alpha) (G) = alpha D (G) + (1- alpha)A (G), where alpha is an element of [0, 1]. Let T-n,T-k be the tree of order n and k pendent vertices obtained from a star K-1,(k) and k pendent paths of almost equal lengths attached to different pendent vertices of K-1,K-k. It is shown that if alpha is an element of [0, 1) and T is a tree of order n with k pendent vertices then rho(alpha)(T) <= rho(alpha) (T-n,T-k), with equality holding if and only if T = T-n,T-k. This result generalizes a theorem of Wu, Xiao and Hong [6] in which the result is proved for the adjacency matrix (alpha = 0). Let q = [n-1/k] and n - 1 = kq + r, 0 <= r <= k - 1. It is also obtained that the spectrum of A(alpha)(T-n,T-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 not equal 0. Thus, the alpha-index of T-n,T-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 not equal 0.