On the Dα-spectral radius of two types of graphs

Let D(G) and T-r(G) be the distance matrix and diagonal matrix with vertex transmissions of a connected graph G, separately. Define matrix D-alpha(G) as D-alpha(G) = alpha T-r(G) + (1 - alpha)D(G), 0 <= alpha <= 1. Let U-n = {G vertical bar G is a simple connected graph with vertical bar V (G)vertical bar = vertical bar E(G)vertical bar = n}, T-n = {T vertical bar T is a tree of order n} and their complement sets be U-n(c) and T-n(c), separately. In this paper, we generalize the conclusions in Qin et al. (2020) to D-alpha-matrix: we depict the extremal graph with maximum D-alpha-spectral radius among U-n(c) (n >= 8) for any alpha is an element of [0, 1/2], and also characterize the graphs among T-n(c) that reach the maximum and minimum of D-alpha-spectral radius for any alpha is an element of [0, 1], respectively. (C) 2020 Elsevier Inc. All rights reserved.